UNIVERSITY  OF  CALIFORNIA 

ANDREW 

SMITH 

HALLIDIC: 


\ 


WORKS  BY  PROF.  DUHEM 

PUBLISHED    BY    A.    HERMANN,    PARIS. 
LE    POTENT1EL   THERMODYNAMIQUE   et  ses   applications  *  la 
mecanique  chimique  et  a  1'etude  des  phe'nomenes  e'lectriques.     i  vol. 
in  8°,  246  pp.     1886. 
HYDRODYNAMIQUE,    ELASTICITE,    ACOUSTIQUE.       2   vol.    in 

lithogr.,  670  pp.     1891-92. 

THEORIE     THERMODYNAMIQUE     DE     LA     VISCOSITE,     DU 
FROTTEMENT    ET  DES    FAUX    EQUILIBRES    CHIMIQUE5. 
i  vol.  gr.  in  8°.     1896. 
TRAITE  ELEMENTAIRE  DE  MECANIQUE  CHIMIQUE,  FONDEE 

SUR  LA  THERMODYNAMIQUE.     4   vol.  gr.   in  8°.     1897-99. 
VOL.    I. — Introduction — Principes    fondamentaux    de  la  Thermodyna- 
mique— Faux  equilibres  et  explosions.     300  pp.     1897. 

VOL.  II. — Vaporisation   et    modifications  analogues — Continuity"   entre 
l'e"tat  liquide  et  1'^tat  gaseux — Dissociation  des  gas  parfaits.    386  pp.    1899. 
VOL.  III. — Les  melanges  homogenes,  les  dissolutions.     400  pp.     1899. 
VOL.  IV. — Les  melanges  doubles — Statique  chimique  ge"ne"rale  des  sys- 
temes  heteVogenes.     384  pp.     1899. 

LES    THEORIES    ELECTRIQUES    DE    J.    CLERK    MAXWELL. 

litude  historique  et  critique,     i  vol.  gr.  in  8°,  235  pp.     1901. 
THERMODYNAMIQUE  ET  CHIMIE.     Le9ons  ^mentaires  k  1'usage 
des  chimistes.     i  vol.  gr.  in  8°,  496  pp.     1902. 

PUBLISHED  BY  A.  GAUTHIER  VILLARS,  PARIS. 
LECONS  SUR  L'ELECTRICITE  ET  LE  MAQNETISME.     3  vol. 

*gr.  i°  8°' 

VOL.  I.— Les  corps  conducteurs  &  I'&at  permanent     560  pp.     1891. 
VOL.  II. — Les  ainaants  et  les  dialectriques.     480  pp.     1892. 
VOL.  III. — Les  courants  line'aires.    528  pp.     1892. 

PUBLISHED  BY  C.  NAUD,  PARIS. 

INTRODUCTION  A  LA  MECANIQUE  CHIMIQUE.  i  vol.  in  8°,  180 
pp.  1893. 

LE  MIXTE  ET  LA  COMBINAISON  CHIMIQUE.  Essai  sur  Evolu- 
tion d'une  idee,  i  vol.  in  8°,  208  pp.  1902. 


'• 

THERMODYNAMICS 

AND 

CHEMISTRY. 


A  NON-MATHEMATICAL  TREATISE  FOR  CHEMISTS 
AND  STUDENTS  OF  CHEMISTRY. 


BY 

P.    DUHEM, 

Correspondant  de  I'Institut  de  France  ;   Professor  of  Theoretical 
Physics  at  the  University  of  Bordeaux. 


AUTHORIZED  TRANSLATION 

BY 

GEORGE  K.  BURGESS, 

Docteur  de  I' Universite  de  Paris;  Instructor  in  Physic*, 
University  of  California. 


FIRST   EDITION. 

FIRST  THOUSAND. 
'''I/O'X 
OF  THE 

UNIVERSITY 

OF 

NEW  YORK : 

JOHN  WILEY  &  SONS. 

LONDON:  CHAPMAN  &  HALL,  LIMITED. 

1903. 


Copyright,  1903, 

BY 

GEORGE  K.  BURGESS. 


Engineering 
Library 


ROBERT  DRUMMOND,  PRINTER,  NEW  YORK. 


AUTHOR'S    INTRODUCTION    TO    AMERICAN 

EDITION. 


MY  DEAR  COLLEAGUE: 

You  wished  that  my  little  book  Thermodynamique  et  Chimie 
appear  in  English,  and  you  have  been  willing  to  undertake  its  trans- 
lation. Both  of  these  determinations  give  me  pleasure. 

I  am  glad,  in  the  first  place,  to  have  my  treatise  rendered  easily 
accessible  to  American  students ;  one  of  the  objects  which  I  had  in 
mind  when  writing  it  was  to  make  the  work  of  J.  Willard  Gibbs 
known  and  admired ;  I  like  to  think  it  will  contribute  to  enhance, 
within  your  active  universities,  the  glory  of  your  illustrious 
countryman. 

Furthermore,  this  glory  is  more  and  more  resplendent  every 
day;  more  and  more  clearly  the  author  of  the  phase  law  appears  as 
the  initiator  of  a  chemical  revolution ;  and  many  do  not  hesitate  to 
compare  the  Yale  College  professor  to  our  Lavoisier. 

Chemists  had  fixed  upon  a  certain  number  of  properties  by 
which  they  recognized  a  substance  to  be  a  definite  compound ;  these 
characteristics  are  effaced  by  the  phase  rule;  many  substances,  to 
which  formulse  had  been  attributed,  are  erased  from  the  number  of 
combinations;  chemical  science  as  a  whole  needs  a  revision  at 
which  the  laboratories  of  America  and  Europe  are  working  most 
diligently. 

Nevertheless,  whatever  be  the  outcome  of  this  revolution,  it 
seems  to  me  there  is  injustice  in  making  the  glory  of  Gibbs  consist 
in  this  alone,  by  seeing  in  him  merely  the  author  of  the  phase  rule. 
In  his  immortal  work.  On  the  Equilibrium  of  Heterogeneous  Substances, 
this  rule  is  not  all;  it  is  but  one  theorem,  and  is  accompanied  by 
other  propositions  whose  importance  is  not  less;  the  theorems  on. 

iii 


139791 


iv      AUTHOR'S  INTRODUCTION   TO  AMERICAN  EDITION. 

indifferent  points,  the  laws  of  dissociation  of  perfect  gases,  the 
properties  of  dilute  solutions,  the  conditions  of  osmotic  equilibrium, 
the  theory  of  the  voltaic  cell,  bear,  not  less  than  the  phase  rule, 
the  mark  of  the  genius  of  their  author. 

The  phase  rule  is  not,  therefore,  by  a  great  deal,  the  whole  of 
Gibbs's  work;  a  fortiori  it  is  not  the  whole  of  chemical  thermo- 
dynamics; other  ideas,  other  principles,  play  an  essential  role  in  the 
development  of  this  science. 

When  a  chemical  system  is  studied,  it  is  assuredly  very  im- 
portant to  determine  its  variance,  whose  value  fixes  the  form  of  the 
equilibrium  law  for  the  system;  but  before  even  calculating  the 
variance,  it  is  expedient  to  answer  this  question :  Is  the  equilibrium 
of  the  system  stable,  indifferent,  or  unstable? 

Thermodynamics  teaches  us  that  no  chemical  equilibrium  is 
unstable.  The  equilibrium  conditions  which  the  chemist  meets 
may  therefore  be  classed  in  two  categories :  those  which  are  stable 
and  those  which  are  indifferent.  This  classification  seems  the  most 
natural,  this  division  the  most  radical  which  may  be  conceived. 
The  systems  in  stable  equilibrium  possess  a  whole  ensemble  of 
properties  which  systems  in  indifferent  equilbirium  do  not  possess; 
the  atter,  in  their  turn,  all  have  certain  properties  which  are  not  met 
when  systems  in  stable  equilibrium  are  studied ;  this  opposition  has 
been  exposed  with  very  great  clearness  by  your  countryman,  Dr. 
Paul  Saurel,  in  the  thesis  he  presented  some  time  since  at  Bor- 
deaux. 

The  equilibria  of  which  I  speak  at  present  are  always  found  at 
the  common  limit  of  two  reactions  opposed  to  each  other;  among 
the  equilibrium  states  studied  by  the  chemist  they  are  only  par- 
ticular cases  or,  better,  limiting  cases.  The  states  of  equilibrium 
with  which  the  chemist  has  actually  to  deal  are  those  which  I  have 
studied  under  the  name  of  false  equilibria.  At  the  beginning  of  his 
immortal  work  Gibbs  has  shown  how  important  is  the  distinction 
between  these  states  and  those  of  which  thermodynamics  treats; 
to  these  last  alone  does  the  phase  rule  apply. 

The  classification,  borrowed  from  the  phase  rule,  of  systems 
into  monovariant,  bivariant,  trivariant,  etc.,  is  therefore  of  extreme 
utility ;  it  arranges  in  admirable  order  a  great  number  of  questions 
in  the  discussion  of  chemical  equilibria;  it  is,  however,  neither  the 


AUTHORS  INTRODUCTION  TO  AMERICAN  EDITION.        v 

only  conceivable  way  of  classifying  these  states  of  equilibrium  nor 
the  most  essential.  This  is  why  I  have  not  thought  it  wise  to  follow 
the  example  given  by  several  recent  treatises  on  chemical  mechanics 
and  base  the  plan  of  my  book  upon  the  phase  rule  alone. 

The  history  of  Gibbs's  work  appears  to  me  to  contain  valuable 
lessons;  the  natural  continuation  of  Lagrange's  Mecanique  analy- 
tique,  it  is  a  powerful  algebraic  attempt  to  express  in  equations  the 
problems  of  Thermodynamics  in  the  most  general  and  abstract  form ; 
but  here  is  this  work  of  a  mathematician  overturning  chemistry! 

This  example  is  quite  fitting  to  indicate  the  close  relationship 
uniting  the  various  sciences,  or,  better,  the  innateness  of  the  separa- 
tions by  which  we  keep  apart  the  various  intellectual  disciplines. 
Being  methods  suitable  to  discover  the  varied  aspects  of  the  truth, 
which  is  single,  they  cannot  be  made  independent  of  one  another; 
whosoever  by  laziness  or  narrow-mindedness  would  pretend  to 
use  but  a  single  one  of  these  methods  would  certainly,  by  isolating 
it,  risk  rendering  it  sterile. 

At  the  start  the  wholly  algebraic  doctrine  of  Gibbs  did  not  find 
in  the  country  of  its  creator  experimentalists  to  transform  it  into 
chemical  theory;  it  first  found  them  in  Holland.  From  this  again, 
it  seems  to  me,  we  may  learn  a  lesson.  The  full  discovery  of  the 
truth  requires  the  concurrence  of  all  peoples,  their  diverse  in- 
tellectual aptitudes,  their  different  ways  of  conceiving  an  idea, 
of  developing  it  and  of  expressing  it.  In  this  respect  exclusive- 
ness  would  again  be  punished  by  sterility. 

You  agree  with  this,  for  you  have  judged  it  useful  to  mitigate 
the  vigorous  and  assertive  initiative  which  the  American  universi- 
ties impart  to  their  students  by  the  discipline  of  equilibrium  and 
of  proportion  which  French  teaching  imposes;  what  you  have  tried 
in  your  own  intellectual  formation  you  wished  to  procure  for 
others ;  you  have  desired  to  make  known  to  your  countrymen  the 
book  in  which  I  have  tried  to  expose,  with  French  neutrality, 
ideas  coming  from  America;  I  could  not  wish  an  interpreter  better 
prepared  to  diffuse  my  thought. 

Believe  me,  my  dear  colleague,  yours  most  sincerely, 

P.  DUHEM. 


PREFACE. 


THE  development  which  Thermodynamics  has  undergone  during 
the  past  fifty  years  merits  the  attention  of  men  engaged  in  the  most 
varied  studies. 

The  opinions,  formerly  admitted  without  protest,  concerning 
the  object  and  influence  of  physical  theories  have  been  overthrown. 
Mechanics  has  ceased  to  be  the  ultimate  explanation  of  the  inorganic 
world ;  it  is  now  but  a  chapter,  the  simplest  and  most  perfect,  of  a 
general  discipline  which  rules  all  the  transformations  of  matter; 
furthermore,  it  is  no  longer  a  question  of  discovering  the  nature 
and  essence  of  these  transformations,  but  merely  of  coordinating 
their  laws  by  means  of  a  small  number  of  fundamental  postulates. 
The  philosopher  follows  with  keen  interest  the  phases  of  this,  one  of 
the  most  considerable  evolutions  which  Cosmology  has  undergone. 

Mathematical  Physics,  at  the  beginning  of  the  nineteenth  cen- 
tury, had  furnished  geometers  with  most  beautiful  and  fruitful 
problems ;  the  efforts  made  to  solve  these  problems  had  given  birth 
to  more  than  one  branch  of  modern  analysis ;  but  it  might  be  feared 
that  the  veins  worked  by  so  many  geniuses  were  exhausted.  The 
new  doctrine  generalizes  very  greatly  the  statements  of  the  prob- 
lems formerly  attacked ;  it  posits  entirely  new  ones,  and  in  this  way 
opens  vast  galleries  to  the  researches  of  the  mathematician. 

The  different  branches  of  physics  seemed  isolated  from  each 
other;  each  of  them  invoked  its  own  principles  and  depended  upon 
particular  methods.  To-day  the  physicist  recognizes  he  is  not 
concerned  with  a  bundle  of  branches  independent  of  each  other,  but 
with  a  tree  whose  branches  start  from  the  same  trunk;  all  the  parts 

vii 


viii  PREFACE. 

of  science  which  he  cultivates  appear  related  as  are  the  members  of 
an  organism. 

Finally,  the  laws  formulated  by  Thermodynamics  impose  a 
rational  order  upon  the  most  confused  chapters  in  Chemistry ;  a  few 
simple,  clear  rules  bring  order  out  of  what  was  a  chaos ;  the  circum- 
stances under  which  the  various  reactions  are  produced,  the  con- 
ditions which  stop  them  and  assure  chemical  equilibrium,  are  fixed 
by  theorems  of  a  geometrical  precision. 

Thus  the  philosopher,  the  mathematician,  the  physicist,  the 
chemist,  are  all  equally  eager  to  know  the  Thermodynamics  of 
to-day,  to  understand  clearly  its  principles,  its  methods,  its  results. 
But  each  of  them  is  interested  in  a  different  aspect  of  this  science; 
a  separate  treatise  is  necessary  for  each. 

It  is  for  the  chemist  these  lessons  are  intended. 

What  the  chemist  expects  above  all  from  Thermodynamics  are 
simple,  clear  rules,  easy  to  use,  which  shall  serve  him  as  con- 
ducting thread  through  the  frightful  labyrinth  of  chemical  facts 
already  known,  which  shall  guide  him  in  the  course  of  his  re- 
searches, which  shall  show  him  exactty,  for  every  reaction,  the 
variable  conditions  at  his  disposal  and  the  essential  conditions 
which  he  is  held  to  determine. 

We  have  done  our  best  to  formulate  these  rules  rigorously  and 
clearly.  We  have  accompanied  each  of  them  with  numerous 
examples ;  in  this  way  we  have  wished  not  merely  to  note  their  im- 
portance and  fruitfulness,  but  also  indicate  the  precautions  to  be 
taken  in  their  application. 

Is  it  sufficient,  however,  for  the  chemist,  to  have  stated  for  him 
the  propositions  to  which  Thermodynamics  leads,  without  analyzing 
the  principles  from  which  they  arise?  Many  think  so;  others  say 
so ;  we  cannot  believe  it. 

Just  as  it  is  unworthy  of  a  man  who  thinks  to  take  certain 
aphorisms  as  guides  for  his  scientific  activity  without  seeking  to 
know  the  titles  these  aphorisms  lay  claim  to,  the  sources  whence 
come  their  authority,  so  this  intellectual  laziness  would  have,  in 
practice,  disastrous  consequences. 

It  is  often  said  there  are  no  rules  without  exception.  Concern- 
ing the  rules  which  Thermodynamics  lays  down  for  Chemical 
Mechanics,  it  would  be  more  exact  to  say  that  every  rule  follows 


PREFACE.  ix 

after  hypotheses,  and  that  no  hypothesis  is  legitimate  outside  of 
certain  precise  and  determined  conditions.  In  Physics  there  is  no 
principle  which  is  true  at  all  times,  in  all  places,  under  every  cir- 
cumstance. Now  the  field  within  which  a  rule  applies  with  secu- 
rity has  for  boundaries  the  limits  of  exactness  of  the  hypotheses 
upon  which  the  rule  depends.  Whoever,  therefore,  does  not  know 
where  a  rule  comes  from  runs  the  risk  of  employing  it  in  cases 
where  its  usage  is  proscribed,  and  of  finding  in  it,  not  a  safe  guide, 
but  a  false  adviser. 

This  is  why,  before  developing  the  laws  of  Chemical  Statics  and 
Dynamics,  we  have  insisted  upon  examining  the  foundations  upon 
which  these  sciences  are  built. 

Our  first  five  chapters  are  devoted  to  this  examination;  we 
have  taken  the  greatest  pains  to  free  the  statements  of  the  primary 
ideas  of  Thermodynamics  from  all  algebraic  complications;  the 
calculus  is  not  used;  in  fact  we  have  supposed  on  the  part  of  the 
reader  no  knowledge  in  mathematics  or  physics  beyond  that  possessed 
by  the  graduate  of  a  good  high  school. 

It  is  by  means  of  the  fundamental  hypotheses  of  algebra  that 
rules  useful  to  the  chemist  are  found;. the  mechanism  of  this  deduc- 
tion cannot,  therefore,  be  separated  from  the  mathematical  formulae 
by  which  alone  it  operates ;  not  wishing  to  write  for  the  geometer, 
we  have  not  deemed  it  necessary  to  analyze  the  parts  of  this  mechan- 
ism ;  but  this  omission  is  hardly  of  any  importance  to  the  chemist ; 
when  the  latter  has  an  exact  knowledge  of  the  conditions  in  which 
it  is  legitimate  to  use  a  principle,  when  he  sees  clearly  the  practical 
consequences  which  are  related  to  this  principle,  he  may,  with 
entire  confidence,  trust  in  the  chain  whose  two  ends  he  holds  in  a 
firm  hand;  for  the  intermediate  links,  which  he  has  not  tested, 
have  the  rigidity  of  algebra. 

Besides,  if  any  one  inclined  and  prepared  for  this  study  de- 
sires to  fill  in  this  chain  and  follow,  in  all  its  developments,  this 
linking  together  of  Chemical  Mechanics,  he  may  easily  satisfy  his 
desire  to  learn ;  we  have  elsewhere  tried  to  help  him  in  this. 

We  have  given  very  considerable  space  to  the  most  recent  appli- 
cations of  Thermodynamics  to  Chemistry.  We  have,  in  particular, 
developed  the  applications  of  that  admirable  phase  law,  an  alge- 
braic theorem  conceived  by  the  genius  of  J.  Willard  Gibbs  and 


X  PREFACE. 

which  the  chiefs  of  the  Dutch  school,  Van  der  Waals,  Bakhuis 
Roozboom,  and  Van't  Hoff,  have  been  able  to  rear  into  one  of  the 
most  precious  guiding  principles  of  modern  Chemistry. 

We  have  studied  with  great  care  those  systems  with  fallacious 
properties  which  have  for  a  long  time  passed  as  definite  com- 
pounds: mixed  crystals,  eutectic  conglomerates,  indifferent  states 
of  double  mixtures.  We  have  neglected  nothing  of  what  may  put 
the  experimenter  on  his  guard  against  these  appearances  of  chemi- 
cal analysis. 

We  have  not  wished,  on  the  other  hand,  that  the  exposition  of 
these  chapters  of  chemical  mechanics,  so  new  and  so  full  of  promise, 
detract  from  the  study  of  the  discoveries  which  have  received  the 
sanction  of  time  and  which  to-day  are  classic.  Disciple  of  Moutier, 
Debray,  Troost,  Hautefeuille,  Gernez.  we  have  not  wished  either 
to  forget  or  let  be  forgotten  that  the  union  of  Thermodynamics 
and  Chemistry  was  accomplished  in  France  in  the  laboratory  of  the 
immortal  Henri  Sainte-Claire  Deville.1 

P.  DUHEM. 

BORDEAUX,  January  2,  1902. 

1  It  was,  in  fact,  the  19th  of  June,  1871,  that  J.  Moutier  communicated  to 
the  Academy  of  Sciences  a  note  entitled  Sur  la  dissociation  au  point  de  vue  de  la 
Thermodynamique  (Comptes  Rendus,  v.  72,  p.  759).  In  this  note  the*  pos- 
sibility of  applying  Clapeyron  and  Clausius'  equation  to  the  dissociation  of 
calcium  carbonate  studied  by  Debray  was  remarked  for  the  first  time.  The 
same  year  Peslin  published,  in  the  Annales  de  Chimie  et  de  Physique  (4th 
Series,  v.  24,  p.  208),  an  article  in  which  Clapeyron  and  Clausius'  equation, 
combined  with  the  measurements  of  dissociation  tensions  given  by  Debray, 
gave  the  numerical  value  of  the  heat  of  formation  of  calcium  carbonate; 
this  value  agreed  with  Favre's  thermochemical  determinations. 

After  these  first  writings,  which  inaugurated  the  union  of  Thermody- 
namics and  Chemistry,  we  have  to  wait  until  1873  to  find,  concerning  this 
question,  new  investigations,  due  in  part  to  J.  Moutier  and  in  part  to  Horst- 
mann  (Liebig's  Annalen  der  Chemie  und  Pharmacie,  v.  170,  p.  192). 


TABLE  OF  CONTENTS. 


PAGE 

INTRODUCTION iii 

PREFACE , vii 

CHAPTER  I. 

WORK  AND  ENERGY 1 

i.  Work  of  a  force  applied  to  a  moving  point,  page  1. — 2.  Appli- 
cation to  the  case  of  weight,  3.— 3.  Work  of  forces  applied  to  a  sys- 
tem, 4. — 4.  Case  of  gravity,  5. — 5.  Case  of  a  system  under  uniform 
pressure,  5. — 6.  Geometrical  representation  of  preceding  results,  7. — 
7.  Application  to  a  gas  that  obeys  Mariotte's  law,  11. — 8.  In  some 
cases,  the  work  of  the  forces  applied  to  a  system  depends  only  on  the 
initial  and  final  states  of  this  system,  12. — 9.  In  general,  the  work 
done  by  the  forces  applied  to  a  system  depends  upon  every  modifica- 
tion the  system  undergoes,  12.— 10.  Potential,  14. — n.  Potential  due 
to  gravity,  15. — 12.  Forces  which  admit  a  potential  in  virtue  of  the 
restrictions  imposed  upon  the  system,  15. — 13.  Energy,  16. — 14.  Prin- 
ciple of  virtual  displacements,  17. — 15.  Conservatives  of  energy. 
Conservative  systems,  18. — 16.  Principle  of  virtual  displacements  for 
conservative  systems.  Stability  of  equilibrium,  19. 

CHAPTER  II. 

QUANTITY  OP  HEAT  AND  INTERNAL  ENERGY 21 

17.  Loss  of  kinetic  energy  and  generation  of  heat  in  a  machine  left 
to  itself,  page  21. — 18.  Mechanical  equivalent  of  heat,  22. — 19.  Prin- 
ciple of  the  equivalence  of  heat  and  work,  22. — 20.  Value  of  the 
mechanical  equivalent  of  heat,  23  — 21.  Extension  of  the  principle  of 
the  equivalence  of  heat  and  work  to  an  unclosed  cycle,  24. — 22.  In- 
ternal energy,  25. — 23.  Principle  of  the  conservation  of  energy,  26  — 

24.  Gases  which  obey  Mariotte's  law.     Absolute  temperature,  27. — 

25.  Expansion  of  a  gas  in  vacuo.     Guy-Lussac's  experiment,  28. — 

26.  Perfect  gases,  30.— 27.  Specific  heat  at  constant  volume.    Inter- 

xi 


xii  TABLE  OF  CONTENTS. 

P.AGB 

nal  energy  of  a  perfect  gas,  30. — 28.  Specific  heat  at  constant  pres- 
sure. Robert  Mayer's  relation,  31. — 29.  Influence  of  temperature  on 
the  specific  heats  of  perfect  gases.  Clausius'  law,  33. — 30.  Evalua- 
tion of  the  mechanical  equivalent  of  heat,  35. 

CHAPTER  III. 

CHEMICAL  CALORIMETRY 36 

31.  The  quantity  of  heat  set  free  by  a  system  which  undergoes  a 
transformation  does  not  depend  solely  upon  the  initial  and  final  states, 
page  36. — 32.  Example  from  the  study  of  perfect  gases,  37. — 33.  Case 
in  which  the  quantity  of  heat  set  free  by  a  system  depends  solely  upon 
the  initial  and  final  states,  38.— 34.  Utility,  in  chemical  calorimetry, 
of  the  preceding  law,  39. — 35.  Exothermic  and  endothermic  reac- 
tions, 41. — 36.  Heats  of  formation  under  constant  pressure  and  at 
constant  volume,  44. — 37.  Case  in  which  the  two  heats  of  formation 
are  equal  to  each  other,  45. — 38.  General  relation  between  the  two 
heats  of  formation,  45. — 39.  Case  in  which  the  compound  is  a  per- 
fect gas,  46. — 40.  The  distinction  between  the  two  heats  of  formation 
has  small  importance  in  practice,  46. — 41.  Influence  of  temperature 
on  the  heats  of  formation,  47.— 42.  Heat  of  formation  referred  to  a 
temperature  at  which  the  reaction  considered  is  impossible,  48. — 
43.  Importance  of  the  variations  that  changes  of  temperature  cause 
in  the  heats  of  formation,  49. — 44.  Case  of  perfect  gases  which  com- 
bine without  condensation.  Delaroche  and  Berard's  law.  The 
heats  of  formation  are  independent  of  the  temperature,  49. 

CHAPTER  IV. 

CHEMICAL  EQUILIBRIUM  AND  THE  REVERSIBLE  TRANSFORMATION 53 

45.  Idea  of  chemical  equilibrium.  It  differs  from  the  idea  of 
mechanical  equilibrium,  page  53. — 46.  The  chemical  equilibrium 
may  be  the  common  limit  of  two  oppositely  directed  reactions.  Phe- 
nomena of  etherification,  53. — 47.  Reciprocal  actions  of  two  soluble 
salts  in  the  midst  of  a  solution,  55. — 48.  Many  chemical  systems 
seem  incapable  of  possessing  a  state  of  equilibrium  which  is  the 
common  limit  of  two  reciprocally  inverse  reactions,  56. — 49.  Grove's 
experiment.  Water  is  decomposable  by  heat,  57. — 50.  Direct  demon- 
stration of  the  dissociation  of  water,  57. — 51.  Dissociation  of  car- 
bonic acid  gas,  59. — 52.  These  decompositions  are  not  complete  but 
limited  at  the  temperatures  at  which  they  are  produced,  the  inverse 
reaction  also  takes  place,  59. — 53.  Example  of  a  state  of  equilibrium 
which  is  the  common  limit  of  two  reactions  the  inverse  of  each  other. 
Action  of  water  vapor  on  iron  and  the  in  verse  action,  61. — 54.  Changes 
of  physical  state  give  rise  to  equilibrium  conditions  of  which  each  is 
the  common  limit  of  two  modifications  the  inverse  of  each  other. 


TABLE  OF  CONTENTS.  xiii 

PAGE 

Saturation  of  solutions,  63. — 55.  Another  example :  Tension  of 
saturated  vapor,  63. — 56.  Dissociation  of  carbonate  of  calcium.  Ten- 
sion of  dissociation,  65. — 57.  The  study  of  chemical  reactions  and 
the  study  of  physical  changes  of  state  depend  on  the  same  theory, 
chemical  mechanics,  67. — 58.  Idea  of  reversible  transformation,  67. 
— 59.  Example  of  reversible  transformation  furnished  by  the  vapori- 
zation of  a  liquid,  70. — 60.  Example  of  reversible  transformation 
furnished  by  the  dissociation  of  cupric  oxide,  72. — 61.  Example  of 
reversible  transformation  furnished  by  the  dissociation  of  water 
vapor,  73. 

CHAPTER  V. 

THE  PRINCIPLES  OP  CHEMICAL  STATICS 75 

62.  The  principle  of  Sadi  Carnot.  Generalization  of  this  principle 
by  Clausius,  page  75.— 63.  The  absolute  temperature  is  always  posi- 
tive, 77. — 64.  Property  of  a  real  isothermal  cycle,  78. — 65.  Impossibil- 
ity of  perpetual  motion,  79. — 66.  Continuously  acting  machines,  80. — 
67.  Statement  of  Carnot  and  Clausius'  principle  for  an  open  rever- 
sible transformation.  Entropy,  80. — 68.  Entropy  of  a  perfect  gas, 
82.— 69.  Statement  of  the  principle  of  Carnot  and  Clausius  for  a 
real  open  transformation.  Compensated  and  non-compensated  trans- 
formations, 85.— 70.  Principle  of  the  increase  in  the  entropy  of  an 
isolated  system,  87. — 71.  Use  of  this  principle  in  chemical  statics, 
88. — 72.  Compensated  and  non-compensated  work  in  an  isothermal 
transformation,  88. — 73.  First  form  of  the  equilibrium  condition  of 
a  system  kept  at  a  given  temperature,  90. — 74.  Expression  of  non- 
compensated  work  accomplished  in  an  isothermal  modification,  91. — 
75.  Characteristic  function  of  a  system,  91. — 76.  Characteristic 
function  of  a  perfect  gas,  92.— 77.  The  characteristic  function  con- 
sidered as  available  energy,  93. — 78.  Definite  form  of  the  equilib- 
rium condition  of  a  system  kept  at  a  given  constant  temperature,  94. 
— 79.  Internal  thermodynamic  potential,  94. — 80.  Total  thermo- 
dynamic  potential  under  constant  pressure  or  at  constant  volume, 
95. — 81.  Stability  of  equilibrium,  96. — 82.  Interpretation  of  the  non- 
compensated  work,  97. — 83.  Intensity  of  reaction;  slow  reactions, 
98.— 84.  Very  intense  reactions;  principle  of  maximum  work,  99. — 
85.  Available  mechanical  effect  of  an  adiabatic  change,  101. — 86.  Ap- 
plication to  the  theory  of  explosives,  103. 

CHAPTER  VI. 

THE  PHASE  RULE 106 

87.  The  number  of  independent  components  of  a  chemical  system 
of  given  kind,  page  106. — 88.  Number  of  phases  into  which  is  divided 
a  given  system,  108. — 89  Fundamental  hypothesis,  108.— 90.  Vari- 


xiv  TABLE  OF  CONTENTS. 

PAGE 

ance  of  a  system,  109.— 91.  Systems  of  negative  variance,  110.— 
92.  Invariant  systems,  110. — 93.  Monovariant  systems.  Transform- 
ation tension  at  a  given  temperature.  Transformation-point  under 
a  given  pressure,  110. — 94.  Examples  of  monovariant  systems,  111.— 
95.  Role  of  monovariant  systems  in  the  history  of  chemical  mechan- 
ics, 113. — 96.  Bivariant  systems,  114.— 97.  Remark  on  the  law  of 
equilibrium  of  bivariant  systems,  114. — 98.  There  are  contradictions 
to  the  phase  rule,  116. — 99.  J.  Moutier's  rule  concerning  these  con- 
tradictions, 117. 

CHAPTER  VII. 

MULTIVARIANT  SYSTEMS 118 

I.  Trivariant  Systems 118 

100.  Multivariant  systems.  Trivariant  systems,  page  118. — 
101.  Theory  of  double  salts,  118. — 102.  Surface  of  solubility  of  a 
double  salt  at  a  given  pressure,  119.— 103.  Case  in  which  the  solution 
may  furnish  two  distinct  salts,  120. — 104.  Conditions  in  which  the 
two  precipitates  are  simultaneously  in  equilibrium  with  the  solution, 
122. — 105.  Case  in  which  the  solution  may  give  three  distinct  salts, 
124. — 106.  The  alloy:  lead,  tin,  bismuth.  Charpy's  researches,  125. 
— 107.  Mixture  of  three  melted  salts,  125. — 108.  Domain  of  a  preci- 
pitate, 126. — 109.  System  water,  ferric  chloride,  hydrochloric  acid. 
Researches  of  Bakhuis  Roozboom  and  Schreinemaker,  127. — 
no.  System  water,  potassium  sulphate,  and  magnesium  sulphate. 
Investigations  of  Van  der  Heide,  127. — in.  System  water,  potassium 
chloride,  and  magnesium  chloride.  Researches  of  Van't  Hoff  and 
Meyerhoffer,  128. 

II.  Quadrivariant  Systems 130 

112.  Quadri variant  systems  formed  of  four  components  divided 

into  two  phases,  page  130. — 113.  Three  salts  dissolved  in  water. 
Solubility  surface  of  a  precipitate  at  a  given  pressure  and  tempera- 
ture, 131 — 114.  System :  water,  magnesium  chloride,  magnesium 
sulphate,  chloride  of  potassium,  potassium  sulphate.  Investigations 
of  Loawenherz,  Van't  Hoff,  and  Meyerhoffer,  133  — 115.  System: 
water,  potassium  chloride,  sodium  chloride,  potassium  sulphate,  sodi- 
um sulphate.  Studies  of  Meyerhoffer  and  Saunders,  138. — 116.  Four 
salts  dissolved  in  water,  one  of  them  to  saturation.  System  :  water, 
sodium  chloride,  potassium  chloride,  sodium  sulphate,  magnesium 
chloride,  140. —117.  Five  salts  dissolved  in  water,  two  of  which  to 
saturation;  a  calcium  salt  added  to  the  preceding  system,  143. 


TABLE  OF  CONTENTS.  XV 

CHAPTER  VIII. 

PAGE 
MONOVABIANT   SYSTEMS 147 

118.  Return  to  monovariant  systems,  page  147. — 119.  A  compo- 
nent existing  in  two  phases,  147. — 120.  Two  components  divided 
among  three  phases,  148. — zaz.  Three  components  with  four  phases, 
148.— 122.  Law  of  equilibrium  for  monovariaut  systems.  Trans- 
formation tension  and  transformation  point,  148. — 123.  Curve  of 
transformation  tensions,  149. — 124.  Curves  of  tensions  of  saturated 
vapor,  149. — 125.  Fusion  phenomena,  149. — 126.  Allotropic  trans- 
formations of  solids,  150. — 127.  Gaseous  substance  and  solid  poly- 
mer, 150.— 128.  Dissociation  tension,  150. — 128.  Transformation 
point  of  double  salts,  167. — 130.  Precautions  to  take  in  the  applica- 
tion of  the  preceding  laws.  First  example:  Dissociation  of  the  red 
oxide  of  mercury.  Pelabon's  investigations,  153. — 131.  Second 
example  :  Dissociation  of  cupric  oxide,  154. — 132.  Dissociation  of 
chlorine  hydrate,  155. — 133.  The  absence  of  a  fixed  tension  of  disso- 
ciation distinguishes  a  solution  from  a  definite  compound,  156. — 
134.  The  zeolites  are  solid  solutions,  157.— 135.  The  existence  of  a 
dissociation  tension  does  not  always  prove  the  existence  of  a  definite 
compound.  Dissociation  of  palladium  hydride,  158. — 136.  Robin's 
law,  162.— 137.  Moutier's  law,  163. — 138.  False  equilibria  in  mono- 
variant  systems,  164. — 139.  Another  form  of  Moutier's  law,  165. — 
140.  Corollary  to  this  law,  165.  — 141.  Consequence  relative  to  very 
low  temperatures;  the  principle  of  maximum  work  is  exact  at  these 
temperatures,  165. — 142.  Consequence  for  high  temperatures,  166. — 
143.  Similarity  of  Moutier's  and  Robin's  laws  Form  of  the  curve  of 
transformation  tensions,  167  —144.  In  every  monovariant  system 
there  may  be  two  modifications,  the  inverse  of  each  other,  which 
change  the  masses  of  the  phases  without  changing  their  composition, 
170.— :i45.  The  equilibrium  of  monovariant  systems  is  indifferent, 
171. — 146.  Law  of  Clapeyron  and  Olau  ius,  171. — 147.  Application 
to  vaporization,  173.— 148.  Appli-ation  to  fusion.  Variation  of 
fusing-point  with  pressure,  173.— 149.  Application  to  the  allotropic 
transformation  of  a  solid  into  another  solid,  175. — 150.  Application 
to  dissociation,  178. 

CHAPTER  IX. 

MULTIPLE  POINTS  OR  TRANSFORMATION  POINTS 180 

151.  The  same  substance  in  the  thre^  states:  liquid,  solid,  gaseous. 
Triple  point,  page  180. — 152.  The  transformation  curve  in  the 
neighborhood  of  the  triple  point,  181. — 153.  Historical,  182. — 154.  Exr 
perimental  verifications,  183. — 155.  Modifications  of  phosphorus. 
Researches  of  Troost  and  Hautefeuille,  184. — 156.  Researches  of 


xvi  TABLE  OF  CONTENTS. 

PAGE 

G.  Lemoine,  186.— 157.  Anomaly  observed  by  Lemoine,  187. — 158. 
Explanation  of  this  anomaly  by  Troost  and  Hautefeuille,  188. — 159. 
The  triple  point  considered  as  transition  point,  189.— 160.  Generaliza- 
tions of  the  preceding  ideas.  Quadruple  points,  192. — 161.  Quad- 
ruple points  in  the  study  of  hydrates  from  a  gas,  193. — 162.  Quintuple 
points,  193. 

CHAPTER  X. 

DISPLACEMENT  OF  EQUILIBRIUM 195 

163.  In  general,  a  modification  which  leaves  invariable  the  compo- 
sition of  each  phase  cannot  be  imposed  upon  a  system  whose  vari- 
ance exceeds  1,  page  195. — 164.  In  general,  the  equilibrium  of  a 
system  whose  variance  exceeds  1  is  stable,  196. — 165.  Displacement 
of  equilibrium  by  variation  of  pressure,  196. — 166.  Various  applica- 
tions, 197. — 167.  Case  of  combination  without  contraction,  198. — 
168.  — Experimental  verifications  :  hydriodic  acid,  198.— 169.  Sele- 
nic  acid,  199. — 170.  Variation  of  the  solubility  of  a  salt  with  pressure, 
200. — 171.  Displacement  of  equilibrium  by  variation  of  the  tempera- 
ture, 202. — 172.  Lowering  of  freezing  points  of  solvents,  203. — 

173.  Lowering  of  the  tension  of  saturated  vapor  of  solvents,  204. — 

174.  Dissociation  of  exothermic  compounds  and  formation  of  endo- 
thermic  compounds  by  rise  in  temperature,  205. — 175.  Actions  pro- 
duced by  a  series  of  electric  sparks;  interpretation  given  by  H.  Saint- 
Claire  Deville.     Apparatus  with  cold  and  hot  tubes,  206. — 176.  Dis- 
sociation   of    carbonous    oxide,    of    sulphurous    and    hydrochloric 
acid  gases.     Synthesis  of  ozone,  208. — 177.  Synthesis  of  acetylene, 
210. — 178.  Case  of  reactions  which  neither  absorb  nor  liberate  heat, 
210.— 179.  Phenomena  of  etherification,  210. — 180.  Minimum  disso- 
ciation of  hydrogen  selenide,  211. — 181.  Similarity  of  the  preceding 
principle  and  Moutier's  law.     At  very  low  temperatures  the  princi- 
ple of  maximum  work  is  exact,  212. 

CHAPTER  XI. 

BIVARIANT  SYSTEMS.    THE  INDIFFERENT  POINT 214 

182.  Various  types  of  bivariant  systems  :  Solutions  and  double 
mixtures,  page  214. — 183.  Law  of  equilibrium  of  bivariant  systems. 
This  equilibrium  is  stable  in  general,  214. — 184.  Solutions.  Satura- 
tion. Solubility  curve,  216. — 185.  For  a  hydrated  salt  two  saturated 
solutions  correspond  to  each  temperature.  The  solubility  curve  has 
two  branches,  216. — 186.  Non-saturated  and  supersaturated  solu- 
tions, 217.— 187.  Heat  of  solutions  in  saturated  solutions,  219. — 
188.  Displacement  of  equilibrium  by  variation  of  temperature,  219. 
— 189.  Precautions  regarding  the  use  of  the  preceding  law,  221. — 
190.  The  two  branches  of  the  solubility  curve  of  a  hydrate  join 


TABLE  OF  CONTENTS.  xvii 

PAGB 

each  other  at  the  indifferent  point  when  the  saturated  solution  has 
the  same  composition  as  the  hydrate,  221. — zgz.  The  temperature  of 
function  is  the  aqueous  fusing  point  of  the  hydrate,  223. — 192.  Ex- 
perimental investigations  of  Guthrie,  Roozboom,  and  other  observers, 
224. — 193.  Indifferent  point  of  a  double  mixture,  227. — 194.  Two 
theorems  of  Gibbs  and  of  Konovalow,  227.— 195.  Application  of  the 
first  theorem  to  mixtures  of  volatile  liquids,  228. — 196.  Distillation 
of  a  mixture  of  two  volatile  liquids  under  constant  pressure,  230. — 
197.  Mixtures  which  pass  over  entirely  by  distillation  without  varia- 
tion of  the  boiling  point,  232.— 198.  These  mixtures  are  not  definite 
compounds.  Researches  of  Roscoe  and  Dittmar,  233. — 199.  Appli- 
cation of  the  second  theorem  of  Gibbs  and  of  Konovalow  to  mixtures 
of  volatile  liquids,  235. — 200.  Distillation  of  a  mixture  of  two  liquids 
at  constant  temperature,  236. — 201.  Relation  between  distillation  at 
constant  pressure  and  distillation  at  constant  temperature,  238. 

CHAPTER  XII. 

BIVARIANT  SYSTEMS  (Continued.)    TRANSITION  AND  EUTEXIA 240 

202.  Common  point  to  the  solubility  curves  of  two  hydrates. 
Three  cases  to  distinguish,  page  240. — 203.  Transition  point,  241. — 
204.  Various  examples.  Sodium  sulphate,  243.— 205.  Thorium  sul- 
phate, 243. — 206.  Eutectic  point,  244. — 207.  Formation  of  the  eutec- 
tic  mixture,  247. — 208.  Particular  case.  Ice  and  anhydrous  salt,  249. 
— 209.  Non-existence  of  cryohydrates,  249. — 210.  Eutectic  point 
between  ferric  chloride  hydrates.  Investigations  of  Bakhuis  Rooz- 
boom, 251. — 2ion.  Hydrates  of  perchloric  acid;  Van  Wyk  investiga- 
tions, 253. — 211.  Researches  of  Van't  Hoff  and  Meyerhoffer  on  mag- 
nesium chloride,  254. — 212.  Roozboom's  researches  on  calcium 
chloride,  256.— 213.  Stortenbeker's  studies  on  iodine  chlorides,  258. 
— 214.  Studies  by  Guthrie  and  by  Le  Chatelier  on  the  mixtures  of 
two  salts,  259. 

CHAPTER  XIII. 

MIXED  CRYSTALS.    ISOMORPHOUS  MIXTURES 262 

215.  Isomorphous  salts;  Riidorffs  observations,  page  262. — 216.  In- 
terpretation of  the  preceding  facts;  isomorphous  mixtures  are  solid 
solutions,  263. — 217.  Theory  of  the  solubility  of  two  isomorphous  salts, 
264.— 218.  Isomorphism  of  the  sulphates  of  the  magnesium  series. 
Studies  of  Stortenbeker,  265. — 219.  Solutions  containing  mixed 
crystals  and  definite  compounds.  Researches  of  Roozboom  and  of 
Retgers,  269. — 220.  Two  melted  isomorphous  salts.  Case  where 
there  is  produced  a  single  kind  of  mixed  crystals,  271. — 221.  Case  in 
which  there  may  be  formed  two  kinds  of  mixed  crystals,  275. — 


xviii  TABLE  OF  CONTENTS. 

PAGE 

222.  The  two  kinds  of  mixed  crystals  maybe  furnished  by  the  liquid 
mixture.  Case  of  the  transition  point,  276. — 223.  Case  of  a  eutectic 
point,  278. — 224.  Isotrimorphous  and  isotetramorphous  substances; 
studies  of  Hissink  and  of  Van  Eyk,  281. — 2240.  Sulphur  and  phos- 
phorus; researches  of  Boulouch,  284. — 2246.  Sulphur  and  selenium; 
W.  E.  Ringer's  researches,  285. 

CHAPTER  XIV. 

MIXED     CRYSTALS     (Continued).      OPTICAL     ANTIPODES,      METALLIC 
ALLOYS 288 

I.  Optical  Antipodes 288 

225.  Mixed  crystals  are  not  limited  to  mixtures  of  isomorphous 
bodies.  Their  frequency  inorganic  chemistry,  page  288. — 226.  Opti- 
cal antipodes.  Inactive  substances  to  which  they  may  give  rise,  289. 
— 227.  Freezing  of  the  mixture  of  two  optical  antipodes,  292. — 
228.  The  congelation  of  the  mixture  furnishes  neither  racemic 
compound  nor  mixed  crystals,  292.— 229.  The  congelation  of  the 
mixture  may  give  a  racemic  compound,  293. — 230.  The  congelation  of 
the  mixture  gives  mixed  crystals,  294. — 231.  Formation  in  solution 
of  a  racemic  compound,  296. 

II.  The  Metallic  Alloys 298 

232.  Liquid  mixtures  which  deposit  metals  in  the  pure  state  or  a 

definite  compound,  page  298.— 233.  Metallic  liquid  mixtures  which 
give  solid  solutions,  301. — 233^.  Zinc  and  antimony  alloys;  Reinder's 
investigations,  303. — 2336.  Cadmium  amalgams,  Byl's  researches, 
305. — 234.  Carburized  iron,  Bakhuis  Roozboom's  theory,  306. 

CHAPTER  XV. 

CRITICAL  STATES 312 

235.  The  critical  point  for  the  vaporization  of  a  single  fluid,  page 
312.— 236.  Double  liquid  mixtures.  The  temperature  at  which  the 
two  layers  have  the  same  composition  does  not  correspond  to  an 
indifferent  point,  314.— 237.  This  temperature  is  a  critical  tempera- 
ture, 316.— 238.  Mixtures  which  separate  into  two  layers  at  tem- 
peratures lower  than  the  critical  temperature,  317. — 239.  Mixtures 
which  separate  into  two  layers  at  temperatures  above  the  critical 
point,  318. — 240.  Influence  of  pressure  on  the  critical  temperature  of 
a  double  liquid  mixture,  319. — 241.  Vaporization  of  a  mixture  of  two 
liquids ;  critical  line  ;.  dew  surface  ;  surface  of  ebullition,  319. — 
242.  Dew  line  and  line  of  ebullition  for  a  mixture  of  given  composi- 
tion, 321. — 243.  Normal  condensation,  retrograde  condensation,  322. 
— 244.  Critical  states  in  the  mixture  of  three  substances,  324. — 
245.  Limiting  crystalline  forms,  325. 


TABLE  OF  CONTENTS.  xix 


CHAPTER  XVI. 

PAGR 

CHEMICAL  MECHANICS  OF  PERFECT  GASES 327 

246.  Necessity  of  new  hypotheses  in  order  to  further  penetrate 
into  the  study  of  chemical  systems,  page  327. — 247.  Properties  of 
the  systems  to  be  studied,  328. — 248.  Hypotheses  which  characterize 
a  mixture  of  perfect  gases,  328. — 249.  Notations,  329. — 250.  Law  of 
equilibrium  of  the  systems  studied,  331. — 251.  Heats  of  reaction  at 
constant  pressure  and  at  constant  volume,  331. — 252.  Tensions  of 
saturated  vapor.  A.  Dupre's  formula,  332. — 253.  Dissociation  ten- 
sions, 334. — 254.  Guldberg  and  Waage's  law,  335. — 255.  Various 
examples  :  ammonium  carbamate,  336. — 256.  Ammonium  cyanide, 
339.— 257.  Influence  of  temperature.  Dissociation  of  mercuric 
oxide,  342.— 258.  Dissociation  of  selenhydric  acid,  344. — 259.  Varia- 
tions in  the  density  of  perchloride  of  phosphorus  vapor,  346. — 
260.  Dissociation  of  a  gas  into  a  vacuum  space,  349.— 261.  Variations 
of  vapor  density.  Are  they  due  to  the  dissociation  of  polymers? 
351. — 262.  Comparison  of  experimental  facts  with  the  theory  of 
dissociation,  353. — 263.  Density  of  iodine  vapor,  354. — 264.  Gibbs's 
formula,  355. 

CHAPTER  XVII. 

CAPILLARY  ACTIONS  AND  APPARENT  FALSE  EQUILIBRIA 357 

265.  The  preceding  theories  are  often  contradicted  by  experiment, 
page  357.— 266.  Rule  stated  by  J.  Moutier  which  summarizes  these 
contradictions,  359. — 267.  True  and  false  equilibria,  359.— 268.  In- 
ternal thermodynamical  potential  of  a  homogeneous  mass  whose 
various  particles  are  infinitely  separated,  359. — 269.  Internal  thermo- 
dynamic  potential  of  a  homogeneous  mass  when  account  is  taken  of 
the  arrangement  of  its  parts,  361. — 270.  Hypothesis  of  molecular 
attraction,  361. — 271.  Internal  potential  of  a  system  divided  into  a 
certain  number  of  homogeneous  phases,  363. — 272.  Comparison 
with  the  form  used  in  the  preceding  chapters,  363.— 273.  When  all 
the  phases  have  very  great  mass,  the  theories  developed  in  the  pre- 
ceding chapters  are  exact,  364. — 274.  Application  to  the  vaporiza- 
tion of  a  fluid;  case  in  which  the  classic  theory  is  exact,  365. — 
275.  Case  where  the  liquid  contains  a  very  small  vapor  bubble. 
Theory  of  retardation  of  ebullition,  365.— 276.  Generalization  of  the 
preceding  considerations,  366. — 277.  Various  phenomena  explained 
by  these  consequences,  367. — 278.  These  phenomena  represent  ap- 
parent false  equilibria,  368. 


xx  TABLE  OF  CONTENTS. 


CHAPTER  XVIII. 

PAGB 

GENUINE  FALSE  EQUILIBRIA 369 

279.  Genuine  false  equilibria  exist.  Investigation  of  H.  Pelabon 
on  the  formation  of  sulphuretted  hydrogen,  page  369. — 280.  The 
condition  of  false  equilibrium  is  not  expressed  by  an  equality,  372. 
— 281.  Region  of  false  equilibria.  Boundary  line  of  false  equilibria, 
372. — 282.  Case  in  which  the  region  of  false  equilibria  separate  two 
regions  corresponding  to  two  reactions  the  inverse  of  each  other. 
"Work  of  Journaux  on  the  reduction  of  silver  chloride  by  hydrogen, 
374. — 283.  Another  example  :  carbonate  of  magnesium  and  bicar- 
bonate of  potassium.  Engel's  studies,  370. — 284.  Return  to  the  idea 
of  reversible  modification,  378. — 285.  Relation  between  the  states  of 
veritable  equilibrium  and  the  states  of  false  equilibrium.  Action  of 
hydrogen  on  silver  chloride  and  the  inverse  action,  379. — 286.  Action 
of  hydrogen  on  selenium  and  the  inverse  action.  Pelabon's  investi- 
gations, 380. — 287.  The  region  of  false  equilibria  separated  from 
that  of  veritable  equilibria  by  a  region  of  unlimited  reaction.  Ac- 
tion of  hydrogen  on  sulphur  and  the  inverse  action,  384.— 288.  Action 
of  oxygen  on  hydrogen.  "Work  of  A.  Gautier  and  H.  Helier,  386. 
— 289.  Action  of  oxygen  on  carbon  dioxide,  388. — 290.  Analogous 
phenomena  shown  by  endothermic  combinations,  389. — 291.  Ozone, 
390. — 292.  Silicon  trichloride.  Investigations  of  Troost  and  Haute- 
feuille,  390.— 293.  Systems  with  unlimited  reaction  and  the  princi- 
ple of  maximum  work,  391. — 294.  Systems  with  unlimited  reaction 
are  not  essentially  distinct  from  systems  with  limited  reaction,  391. 
— 295.  One  may  always  cool  a  chemical  system  sufficiently  for  it  to 
exist  in  the  state  of  false  equilibrium,  393. — 296.  False  equilibria  at 
very  low  temperatures.  Pictet's  researches,  393. — 297.  The  reaction 
point,  394. — 298.  Reaction  point  in  the  phosphorescence  of  phos- 
phorus, 396. — 299.  Analogy  of  the  states  of  false  equilibria  with  the 
mechanical  equilibria  due  to  friction,  398. — 300.  The  existence  of 
false  equilibria  in  chemical  systems  is  not  exceptional  but  regular, 
400. 


CHAPTER  XIX. 

UNEQUALLY  HEATED  SPACES 403 

301.  Formation  and  dissociation  of  selenhydric  acid  in  an  unequally 
heated  space.  Three  cases  to  distinguish,  page  403. — 302.  Phe- 
nomena of  apparent  volatilization,  409. — 303.  Vaporization  presents 
phenomena  analogous  to  those  just  studied,  410. 


TABLE  OF  CONTENTS.  xxi 


CHAPTER  XX. 

PAGE 

CHEMICAL  DYNAMICS  AND  EXPLOSIONS 412 

304.  Chemical  dynamics,  page  412. — 305.  Velocity  of  a  reaction, 
412. — 306.  Fundamental  principle  of  chemical  dynamics,  413. — 
307.  Acceleration  of  a  reaction,  413. — 308.  Comparison  of  the  funda- 
mental principle  of  chemical  dynamics  and  the  principle  of  dynamics 
properly  so  called,  413. — 309.  Influence  of  the  composition  of  the 
system  on  the  velocity  of  reaction,  415. — 310.  Every  isothermal  reac- 
tion is  a  moderated  reaction,  415. — 311.  The  acceleration  of  a  mod- 
erated reaction  is  negative,  416. — 312.  Influence  of  temperature  on 
the  velocity  of  reaction,  416. — 313.  Example  :  Phenomena  of  etherifi- 
cation,  416. — 314.  Variation  of  the  velocity  due  to  a  small  change  of 
composition  and  temperature,  418. — 315.  Return  to  isothermal  reac- 
tions, 418. — 316.  Adiabatic  reactions,  418. — 317.  An  adiabatic  reaction 
may  have  a  positive  acceleration,  420. — 318.  Reactions  with  positive 
acceleration  and  explosive  reactions,  420.— 319.  Conditions  in  order 
that  an  adiabatic  reaction  be  explosive,  421. — 320.  Indetermination 
of  the  temperature  which  renders  a  reaction  explosive,  421. — 321.  Sta- 
bility and  instability  of  limiting  false  equilibria,  421. — 322.  Every 
state  of  false  equilibrium  which  is  not  limited  is  indifferent,  423.— 
323.  If  the  temperature  is  constant,  every  false  equilibrium  is  stable, 
423.— 324.  If  the  reactions  are  all  adiabatic,  the  limiting  false  equi- 
libria may  be  stable  or  unstable,  424.— 325.  Relation  between  the 
limiting  false  equilibria  which  are  unstable  and  the  explosive  reac- 
tions, 427. — 326.  Three  cases  to  distinguish,  427.— 327.  The  reaction 
point  of  a  mixture  is,  in  general,  below  the  explosion  point,  429. — 
328.  The  interval  between  these  two  points  and  the  safety  explosives, 
430. — 329.  Mixtures  which  are  never  detonating,  430.— 330.  Explo- 
sive combinations,  430.— 331.  Influence  of  pressure  on  the  point  of 
explosion,  431. 

LIST  OF  AUTHORS  CITED  IN  THIS  WORK 435 

INDEX  OF  CHEMICAL  SUBSTANCES  STUDIED  IN  THIS  WORK 439 

GENERAL  INDEX 443 


THERMODYNAMICS  AND  CHEMISTRY. 


CHAPTER  I. 
WORK  AND  ENERGY. 

i.  Work  of  a  force  applied  to  a  movable  point. — An  elementary 
definition  of  work  is  given  when  the  following  conditions  are  ful- 
filled: 

Under  the  action  of  a  force  F,  constant  in  magnitude  and  direc- 
tion (Fig.  1),  a  material  point  moves  a  ======= ^ 

length    MM'  =  l    in    the    direction    of    the M  ™ 

,  FIG.  1. 

force. 

We  call  the  work  done  by  the  force  F  the  product  Fl,  with  the 
4-  sign  if  the  material  point  moves  in  the  direction  of  the  force, 
with  the  —  sign  if  the  material  point  moves  in  the  opposite  direc- 
tion. 

This  definition  is  sufficient  to  fix  the  unit  of  work.  When  a 
material  point  under  the  action  of  a  constant  force  equal  to  the 
unit  of  force  moves,  in  the  direction  of  the  force,  a  unit's  distance, 
the  work  done  is  equal  to  unity. 

In  the  C.G.S.  system,  in  which  the  unit  of  length  is  the  centi- 
metre and  the  unit  of  force  the  dyne,  the  unit  of  work  is  the  dyne- 
centimetre  or  erg. 

The  preceding  definition  may  be  so  generalized  that  it  applies  to 
the  case  in  which  under  the  action  of  a  force  F,  constant  in  mag- 
nitude and  direction  (Fig.  2),  a  material  point  moves  a  length 
MM'  =  l  along  a  straight  line  which  makes  a  certain  angle  with 
the  direction  of  the  force.  In  this  case,  the  force  F  is  projected 


2 


THERMODYNAMICS  AND  CHEMISTRY. 


M 


ivr 

FIG.  2. 


upon  the  path  of  the  material  point;   let  /  be  such  projection. 

The  work  of  the  force  F  is  then  the  product 
^  fl,  either  +  or  —  in  sign  according  as  the  point 

moves  in  the  d.rection   of  the  force  /  or  in  the 

opposite  sense. 

Let   a    be    the   angle    (less    than  two  right 

angles)  that  the  direction  of  the  force  F  makes 

with  the  direction  MM'  of  the  displacement  of 
the  material  point;  by  the  definition  of  the  cosine  of  an  angle, 
the  work  we  have  just  described  will  be  represented  in  magnitude 
and  direction  by  the  formula  /  > 

(1)  W=lFcosa. 

This  formula  may  be  interpreted  in  another  way.  I  cos  a  rep- 
resents the  projection  of  the  d  splacement  MM'  on  the  direction 
of  the  force  F,  this  projection  having  its  proper  sign  according  to 
the  ordinary  convention,  that  is,  counted  positive  or  negative  ac- 
cording as  it  is  in  the  direction  of  the  force  F  or  oppositely 
directed.  We  may  then  say  that  when  a  point,  under  the 
action  of  a  force  constant  in  magnitude  and  direction,  moves  in  a 
straight  line,  the  work  of  the  force  is,  in  magnitude  and  direction, 
the  product  of  the  magnitude  of  the  force  by  the  projection  of  the 
displacement  of  the  material  point  on  the  direction  of  the  force. 

_Suppose  that  a  material  point  describes  a  rectilinear  segment 
MM'  =  l  (Fig.  3)  under  the  action  of  the  force  F  which  rests  con- 
stant in  magnitude  and  direction 
during  this  displacement;  that  it 
then  describes  a  second  rectilinear 
segment  M'M"=  '  under  the  action 
of  another  force  F']  then  a  third 
segment  M"M'"  =  l"  under  the 
action  of  a  third  force  F",  and  so 
on.  The  work  done  while  the  ma- 
terial point  describes  the  broken 
path  MM'M"M'"  ...  is,  by  defi- 


F' 


FIG.  3. 


nition,  the  sum  of  the  work  done  during  the  separate  rectilinear 
components  of  the  broken  path;  this  work  has  then  the  value 


(2) 


W = Fl  cos  a  +  F'V  cos  a!  +  F"l"  cos  a."  + 


WORK  AND  ENERGY.  3 

Consider  now  a  material  point  which  describes  a  curved  path 
MN  (Fig.  4),  while  the  force  F 
acting  upon  it  changes  continually 
in  magnitude  and  direction. 

Within  the  curve  MN  inscribe 
a  rectilinear  path  MM'M"M"' 
.  .  .  N.  Let  F,  F',  F",  Fr"  be 
the  values  of  the  variable  force 
at  the  instants  the  material  po\nt 
occupies  the  positions  M,  M',  M", 
M"f.  Let  a  be  the  angle  of  the 
two  directions  F  and  MM',  a' 
the  angle  of  the  two  directions 
F'  and  M'M",  and  so  on. 

If  the  material  point  considered  describes  the  rectilinear  seg- 
ment MM'  under  the  action  of  the  constant  force  F,  the  seg- 
ment M'M"  under  the  action  of  the  force  F',  the  segment  M"M'" 
under  the  action  of  the  force  F",  finally  the  segment  M'"N  under 
the  action  of  the  force  F"',  the  work  done  wiU  have,  according  to 
formula  (2),  the  value 

Fl  cos  a  +  F'l'  cos  a'  +  F"l"  cos  a"  +  F'"lr"  cos  a'". 

Suppose  now  that  we  increase  indefinitely  the  number  of  points 
of  division  M',  M",  M'",  .  .  .  marked  on  the  curve  MN,  causing 
each  of  the  segments  MM',  M'M",  M"M'",  ...  to  approach 
zero.  The  broken  path  MM'M"M'"  .  .  .  will  approach  the 
curved  path  MN.  At  the  same  time  the  preceding  sum  will 
approach  a  limit.  This  limit  is,  by  definition,  the  work  done  while 
the  material  point  describes  the  curved  path  MN. 

2.  Application  to  the  case  of  weight. — Let  us  apply  these  defi- 
nitions to  the  very  simple  case  of  a  movable  point  under  the  action 
of  its  weight. 

Let  M  be  a  material  point;  if  we  denote  its  mass  by  m  and  the 
intensity  of  gravity  by  g,  its  weight  P  is  a  constant,  vertical  force, 
directed  downwards  and  of  magnitude  P=mg. 

Suppose  that  under  the  action  of  its  weight  the  point  describes 
the  rectilinear  path  MM'  (Fig.  5).  Let  z  be  the  distance  of  the 
point  M,  and  &  the  distance  of  the  point  M',  above  an  arbitrary 


4  THERMODYNAMICS  AND  CHEMISTRY. 

horizontal  plane  H]  it  is  clear  that  the  projection  MN  of  the 
path  MM'  on  the  direction  of  the  force  P,  that  is  on  the  vertical 
directed  downwards,  is  represented  in  magnitude  and  direction  by 
the  difference  («  —  «')•  Tne  work  done  during  the  rectilinear  dis- 
placement MM'  of  the  material  point  has  then  for  value 


It  is  equal  to  the  product  of  the  weight  of  the  material  point 
and  the  height  of  the  fall. 


M 


FIG.  5. 


FIG.  6. 


Consider  now  a  point  having  mass  which  moves  along  a  broken 
line;  by  definition  the  work  done  will  be  the  sum  of  the  products 
obtained  by  multiplying  the  weight  of  the  material  point  by  the 
height  of  the  partial  fall  relative  to  each  of  the  rectilinear  segments 
which  compose  the  broken  trajectory;  it  is  then  clear  that  the 
work  done  when  a  material  point  describes  a  broken  path  is  obtained 
by  multiplying  the  weight  of  the  material  point  by  the  height  of 
the  total  fall. 

By  the  method  of  limits,  this  result  may  be  immediately  ex- 
tended to  the  case  of  a  material  point  which  describes  any  curved 
path  whatever  M0Ml  (Fig.  6).  Let  z0  be  the  initial  distance  M0ra0, 
and  ZL  the  final  distance  Mjnv  of  this  point  above  an  arbitrary 
horizontal  plane  H;  the  work  done  by  gravity  during  the  displace- 
ment of  the  point  will  have  the  value 

(3)  W =  mg(z()  —  zi). 

3.  Work  of  forces  applied  to  a  system. — When  a  mechanical 
system,  composed  of  material  points,  is  displaced  and  deformed 


WORK  AND  ENERGY.  5 

under  the  action  of  forces  which  act  on  these  various  points,  the 
work  done  is,  by  definition,  the  sum  of  the  work  done  in  the  dis- 
placements of  the  various  material  points. 

4.  Case  of  gravity.—  Let  us  apply  this  definition  to  the  work 
done  when  a  system  of  material  points  is  deformed  and  displaced 
under  the  action  of  gravity. 

Let  m,  m',  m"  .  .  .  be  the  masses  of  the  various  material 
points  composing  the  system;  z0,  zj,  z9"  ...  the  distances  of  the 
various  points  above  an  arbitrary  horizontal  plane  H,  at  the 
beginning  of  the  displacement. 

In  virtue  of  the  equality  (3)  and  of  the  preceding  definition, 
the  work  done  will  have  for  value 


But  mgz0+m'gz0'  +m"gz  "+  ...  is  the  sum  of  the  moments, 
with  respect  to  the  plane  H  of  the  weights  of  the  various  material 
points,  taken  in  their  initial  positions;  it  can  be  shown  that  this 
sum  is  equal  to  the  moment  with  respect  to  the  plane  H  of  the 
total  weight  of  the  system,  this  weight  being  applied  at  the  initial 
position  of  the  centre  of  gravity. 

If,  then,  we  write  M=m+m'  +  m"+  ...  for  the  total  mass 
of  the  system,  and  Z0  for  the  initial  distance  of  the  centre  of  grav- 
ity of  the  system  above  the  horizontal  plane  H,  we  have 


mgz 

Whence  it  follows  that 
(4)  W=Mg(Z0-Zl). 

When  a  material  system  having  weight  is  deformed  and  displaced 
in  any  manner  whatever  due  to  its  weight,  the  work  done  is  the 
product  of  the  weight  of  the  system  by  the  height  of  the  fall  of  the  centre 
of  gravity. 

5.  Case  of  a  system  under  uniform  pressure.  —  Take  another 
example  where  it  will  be  easy  to  calculate  the  work  done  in  a 
deformation  of  the  system. 

A  reservoir  R  (Fig.  7)  contains  a  gas;  a  pipe,  supposed  cylin- 
drical, is  soldered  to  this  reservoir;  the  pipe  encloses  a  liquid  which 
exerts  a  certain  pressure  on  the  gas;  S  is  the  level  of  the  liquid; 
at  all  points  of  the  surface  S  the  pressure  exerted  on  the  gas  is 


6 


THERMODYNAMICS  AND  CHEMISTRY. 


vertical,  directed  upwards,  and  has  the  same  value  P;  we  will 
suppose  that  this  value  stays  constant,  while  the  level  of  the 
liquid  falls  from  S  to  S',  and  we  shall  calculate  the  work  done. 

Take  on  the  surface  S  a  very  small  area  s;  it  is  acted  upon  by 
a  vertical  force,  directed  upwards,  of  value  Ps.  In  the  displace- 
ment of  the  surface  S  the  various  points  of  the  area  s  follow 
paths  parallel  to  the  generatrices  of  the  cylinder,  so  that  the  sur- 


FIG.  7. 

face  s  is  brought  to  s';  the  projection  of  a  point  in  the  surface  s' 
on  the  direction  of  the  displacement  force  will  be,  in  magnitude 
and  direction,  equal  to  —h,  designating  by  h  the  height  of  the 
cylinder  included  between  S  and  S'.  The  force  applied  to  the  sur- 
face s  does  work  equal  to  —  Phs.  To  obtain  the  total  work  it  will 
be  necessary  to  compute  the  separate  work  done  on  the  areas  s 
into  which  the  area  S  may  be  decomposed  and  find  the  sum  of 
these  work  components.  Now,  in  the  various  terms  of  this  sum, 

—  Ph  will  be  a  common  factor;   noting  that  the  sum  of  the  areas 
such  as  s  is  equal  to  the  area  S,  we  see  that  the  work  sought  is 

—  PhS.     But  hS  is  the  volume  of  the  cylinder  included  between 
the  surfaces  S  and  S';    it  is  thus  the  increase  undergone,  during 
the  modification  considered,  by  the  volume  of  the  gas;  if  V0  is  the 
initial  value  of  this  volume  and  Vl  the  final  value,  we  have  evi- 
dently hS=  Vl  —  V0,  and  the  work  sought  has  the  value 


WORK  AND  ENERGY.  7 

This  is  the  product  of  the  pressure  on  the  gas  and  the  decrease 
(or  increase  with  sign  changed)  of  the  volume  that  it  occupies. 

The  result  that  we  have  just  found  may  be  generalized  in  a 
manner  that  we  shall  state  without  demonstration. 

Every  time  that  a  body  passes  from  volume  V0  to  the  volume 
Fj,  while  the  surface  sustains  a  uniform  normal  pressure  of  con- 
stant value  P,  the  work  done  has  the  value 

(5)  W=P(VQ-VJ. 

Suppose  now  that  a  body  changes  its  volume  while  it  sustains 
a  pressure  which  is  again  normal  and  uniform,  but  whose  magni- 
tude varies  in  the  following  manner : 

While  the  volume  of  the  body  passes  from  the  value  V  to  the 
value  V,  the  pressure  keeps  the  invariable  value  P;  it  has  the 
value  P'  while  the  volume  changes  from  V  to  F";  the  value  P" 
while  the  volume  changes  from  V"  to  V",  and  so  on.  The  work 
done  during  such  a  transformation  is  the  sum  of  the  work  done 
during  the  partial  transformations,  each  of  which  is  performed 
under  constant  pressure;  it  has  for  value 

(6)  W=P(V-V')+P'(V'-V")+P"(V"-V'")  +  .  .  . 

This  work  is  obtained  by  multiplying  each  of  the  diminutions  of 
volume  sustained  in  the  system  by  the  constant  pressure  that  it  sup- 
ports during  this  diminution  of  volume  and  taking  the  sum  of  the 
products  thus  obtained. 

By  the  principle  of  limits  this  method  may  be  extended  to  the 
case  in  which  the  value  of  the  pressure  varies  in  a  continuous 
manner  during  the  change  in  volume  which  the  system  undergoes. 

6.  Geometrical  representation  of  preceding  results.— Take  a 
right  angle  VOP  (Fig.  8)  whose  sides  0V,  OP  are  coordinate  axes 
and  their  intersection  0  the  origin  of  coordinates.  One  of  the  axes 
0V  is  considered  as  having  the  direction  from  left  to  right;  the 
other  axis  OP  is  directed  vertically  upwards;  the  first  is  called  the 
axis  of  abscissce  and  the  second  the  axis  of  ordinates. 

Consider  a  system  occupying  the  volume  V  under  the  pressure 
P.  Along  the  axis  of  abscissae  lay  off  from  the  point  0  a  length 
0V,  measured  by  the  member  F;  along  the  axis  of  ordinates  lay 
off  from  0  a  length  OP,  measured  by  the  member  P;  through  the 


8 


THERMODYNAMICS  AND  CHEMISTRY. 


point  V  draw  a  parallel  to  OP  and  through  P  a  parallel  to  OF; 
these  two  lines  meet  in  a  point  M  which  represents  the  system; 
this  point  has  for  abscissa  the  member  V  and  for  ordinate  the 
member  P;  these  two  members  are  the  coordinates  of  the  point. 

Suppose  that,  under  a  pressure  which  remains  fixed,  the  volume 
of  a  system  passes  from  the  value  V0  to  the  less  value  Fr  The 
point  representing  the  system,  whose  ordinate  keeps  the  invari- 
able value  P,  will  describe  (Fig.  9)  a  straight  line  parallel  to  OF 


M, 


FIG.  8. 


V,  V0 

FIG.  9. 


from  right  to  left  starting  from  the  point  M0,  abscissa  F0,  to  the 
point  Mv  abscissa  Ft.  The  work  done,  according  to  equation  (5), 
will  be  W=P(VQ—Vl),  so  that  we  shall  have,  in  magnitude  and 
direction, 

TF=area 


Suppose,  again,  that  the  volume  of  the  system  diminishes  con- 
tinuously from  F0  to  Vt;  but  let  us  not  suppose  that  this  diminu- 
tion takes  place  under  constant  pressure;  .imagine  that  the  volume 
decreases  from  F0  to  V  under  the  constant  pressure  P,  from  V 
to  V"  under  the  constant  pressure  P',  from  V"  to  V"  under  the 
constant  pressure  P",  finally  from  V"  to  Vl  under  the  constant 
pressure  P'"  .  The  point  representing  the  system  (Fig.  10)  will 
describe  the  broken  line  Mjn'M'M"m"M"'m"'Mt.  The  work 
done  is,  by  formula  (6), 

TF=P(F0-  V)  +P'(F'-  F")  +  P"(F"-  F'" 


This  equality  may  also  be  written 

TF=area  M0m'M'  .  .  .  M'"m'"M1V1V0M 


WORK  AND  ENERGY. 


9 


Suppose  finally  that  each  of  the  diminutions  of  volume  under- 
gone by  the  system  between  the  extreme  values  F0,  Ft  becomes 
smaller  and  smaller;  that  these  diminutions  are  more  and  more 
numerous;  that  from  one  of  these  diminutions  of  volume  to  the 
next  the  pressure  changes  less  and  less :  the  volume  of  the  system 
will  tend  to  diminish  in  a  continuous  manner;  the  broken  path 
MQm'Mf  .  .  .  Mlf  described  by  the  representative  point  (Fig.  11), 
will  approach  the  curved  path  M0M'  .  .  .  M1}  drawn  from  right 


p 

D' 

P 

A"            1 

A' 

p" 

f 

A'"    Tit' 

' 

mf      M, 

p 

o"' 

Mj 

m'" 
"* 

-*  —  ! 

p 

i 

i 

i 
i 
• 

i 

Vl      V" 


v,; 


V"  V  V0 


FIG.  10. 


FIG.  11. 


to  left.  From  Art.  5  it  follows  that  the  work  done  during  this 
continuous  transformation  will  be  represented  by  the  area  bounded 
in  part  by  the  curve  traced: 


(7) 


W  =  area 


Instead  of  decreasing  indefinitely,  the  volume  of  the  system 
might  increase  indefinitely  from  the  value  V0  to  the  greater  value 
Vlf  while  the  pressure  varies  in  a  continuous  manner;  the  point 
representing  the  system  (Fig.  12)  would  describe  from  left  to  right 
the  curved  line  MQMr  A  series  of  considerations  analogous  to 
the  preceding  would  show  that  the  work  done  has  for  value 

(7')  W  =  -area 


Thus,  when  the  volume  of  the  system  has  a  single  direction  of 
variation  between  the  two  extreme  values  V0  and  Vlt  the  area 


gives  always  the  absolute  value  of  the  work  done; 
but  this  absolute  value  must  have  the  sign  -f  or  —  according  as 
the  representative  point  describes  the  curve  MQMl  from  right  to 
left  or  left  to  right. 


10 


THERMODYNAMICS  AND  CHEMISTRY. 


The  volume  of  a  system  which  is  being  modified  may  not  always 
change  in  the  same  sense;  it  may,  for  example,  decrease  at  first 
from  V0  to  Vv  then  increase  from  Vl  to  V2,  and  decrease  finally 
from  V2  to  V3.  The  representative  point  (Fig.  13)  will  first  de- 
scribe the  curve  MQM^  from  right  to  left,  then  the  curve  M1M2 
from  left  to  right,  and  finally  the  curve  M2M3  from  right  to  left, 


O    V0 


FIG.  12. 


the  points  M0,  Mlr  M2,  M3  having  respectively  for  abscissae  the 
numbers  V0)  Vlf  V2,  V3. 

To  each  of  the  three  partial  transformations  where  the  volume 
has  a  single  direction  of  variation,  one  or  the  other  of  the  two  pre- 
ceding theorems  is  applicable;  be:ides,  the  work  done  in  the  total 
transformation  is  the  sum  of  the  work  done  in  each  of  the  three 
partial  transformations;  this  work  has  then  the  value 

TF=area  MQM,V,VQMQ 
-area  Af1M3F2F1M1 
+area  M2M3V3V2M2. 

Evidently  an  evaluation  of  this  kind  will  always  be  applicable, 
even  for  the  most  complicated  cases. 

Let  us  apply  this  mode  of  evaluation  to  the  case  in  which  the 
system  occupying  in  the  first  place  the  volume  V0  under  the  pres- 
sure P  undergoes  a  modification  which  brings  it  back  to  the  same 
volume  and  pressure. 

Suppose,  as  in  Fig.  14,  that  the  curve  consists  of  a  single  loop 


WORK  AND  ENERGY. 


11 


described  by  the  representative  point  in  a  direction  opposite  to  that 
of  the  motions  of  the  hands  of  a  watch.     We  shall  have,  by  equation 

(8), 


—area 

+area  M2MQV0V2M2, 


M 


V,       V0  V2  V 

FIG.  14. 


O       V2 


FIG.  15.    . 

or,  in  uniting  the  two  areas  having  the  sign  +  , 
TF=area 


or  finally 

(9)  TF=area  included  by  the  closed  curve. 

If,  as  in  Fig.  15,  the  closed  path  of  the  representative  point  is 
composed  of  a  single  loop  described  in  the  direction  of  the  motion 
of  the  hands  of  a  watch,  we  find  in  an  analogous  manner  that  the 
work  done  has  the  value 


(9') 


W=  —area  included  by  the  closed  curve. 


7.  Application  to  a  gas  that  obeys  Mariotte's  Law. — Suppose 
we  compress  from  volume  V0  to  the  less  volume  Vl  a  mass  of  gas 
whose  temperature  is  kept  constant,  and  that  the  gas  obeys  Mari- 
otte's Law. 

Let  P0  be  the  pressure  supported  by  this  gas  at  the  moment 
it  occupies  the  volume  F0;  when  it  occupies  the  volume  V,  in- 


12 


THERMODYNAMICS  AND  CHEMISTRY. 


eluded  between  F0  and  V ,  it  will  support  a  pressure  P  given  by 
the  equation 

P  V 
(10)  P^^-. 


P0 


This  equation  gives  the  ordinate  of  the  representative  point  M 

when  V  is  the  abscissa  (Fig.  16);  we 
notice  that  this  ordinate  is  the 
greater  as  V  diminishes;  it  increases 
beyond  limit  when  F0  tends  towards 
zero  and  approaches  zero  when  V 
increases  indefinitely.  The  locus  of 
the  point  M  is  what  the  geometers 
call  a  branch  of  an  equilateral  hyper- 
bola having  for  asymptotes  the  two 
axes  OP  and  0V. 

From   what   we    have    seen    in 
v    Art.  6,  the  work  done  by  the  pres- 
sure  when   the   gas   is   compressed 
to  the  volume  Fx  is  measured  by  the  area 


o    v 


.        V  V0 

RIG.  16. 

from  the  volume  F0 
bounded  by  the  four  following  lines: 

1°.  The  arc   M0Ml  of  the  equilateral  hyperbola   defined  by 
equation  (10); 

2°  and  3°.  The  two  lines  F0M0,  FjMj,  parallel  to  the  axis  of 
ordinates; 

4°.  The  part  F0Fj  of  the  axis  of  abscissae. 

This  area  may  be  found  by  geometry  and  has  the  value 


(ID 


Vt 


the  symbol  log  designating  a  logarithm  in  Briggs's  tables. 

8.  In  some  cases,  the  work  of  the  forces  applied  to  a  system 
depends  only  on  the  initial  and  final  states  of  this  system.— The 

comparison  of  equation  (4),  which  gives  the  work  due  to  gravity, 
with  equation  (6),  which  gives  the  work  of  a  normal  and  uniform 
pressure,  leads  to  one  of  the  most  important  principles  in  mechanics. 
In  .order  to  know  the  work  done  by  gravity  acting  on  a  system 
of  given  mass,  it  is  sufficient  to  know  the  initial  and  final  height 


WORK  AND  ENERGY.  13 

of  the  centre  of  gravity;  it  is  quite  superfluous  to  know  the  form 
of  the  path  the  body  has  taken  in  its  fall  and  the  changes  of  shape 
or  of  dimensions  that  it  may  have  undergone  during  this  fall. 
The  work  done  by  gravity  depends  exclusively  on  the  initial  and  final 
states  of  the  system  on  which  it  acts. 

9.  In  general,  the  work  done  by  the  forces  applied  to  a  sys- 
tem depends  upon  every  modification  the  system  undergoes.  — 
It  is  not  true,  as  in  the  case  of  weight,  that  the  principle  an- 
nounced in  Art.  8  holds  for  the  work  done  by  any  force. 

Imagine  that  a  system,  under  a  uniform,  normal  pressure, 
passes  from  the  initial  state  in  which  it  occupies  the  volume  VQ 
at  the  pressure  P0  to  a  final  state  in  which  it  occupies  the  volume 
Vl  at  the  pressure  Pr 

From  the  initial  to  the  final  state  it  may  pass  in  an  infinitely 
great  number  of  ways;  let  us  choose  two  and  calculate  for  each 
the  work  done  by  the  pressure: 

1°.  The  system  expands,  at  constant  pressure  P0,  from  the 
volume  V0  to  the  volume  V^;  then,  without  change  of  volume, 
change  the  pressure  from  the  value  P0  to  the  value  Pt;  suitable 
changes  of  temperature  allow  accomplishing  these  two  operations; 
during  the  first  part  of  the  transformation  the  pressure  does  work 
equal  to  P^V^-V^,  by  equation  (5);  during  the  second  part  the 
representative  point  describes  a  straight  line  parallel  to  OP,  so 
that  no  work  is  done.  Work  done  by  the  pressure,  in  the  modifi- 
cation considered,  has  the  value 


2°.  Under  the  constant  volume  V0  the  pressure  changes  from 
P0  to  Pj;  then,  at  the  constant  pressure  P1;  the  volume  changes 
from  T0  to  V^,  during  the  first  part  of  the  modification  the  pres- 
sure does  no  work;  the  work  done  during  the  whole  modification 
reduces  to  that  done  during  the  second  part  and,  from  equation 
(5),  has  the  value 


Since  Pl  and  P0  are  different,  the  work  W2  done  during  the 
second  modification  is  not  equal  to  Wv  the  work  done  during  the 
first. 


14  THERMODYNAMICS  AND  CHEMISTRY. 

Suppose  that  the  system  is  brought,  in  the  initial  and  final 
conditions,  to  the  same  temperature;  we  may  make  it  pass  from  the 
first  to  the  second  without  variation  of  temperature;  this  third 
modification  will  differ,  in  general,  from  the  two  preceding;  if 
the  system  is  a  gas  that  obeys  Mariotte's  Law,  we  may  find  the 
value  of  the  work  done  during  this  last  modification  ;  this  value, 
given  by  equation  (11),  will  differ  again  from  the  two  preceding. 

Thus,  the  value  of  the  work  done  when  a  system,  supporting  a 
uniform,  normal  pressure  at  every  point  of  its  surface,  passes  from  a 
given  initial  state  to  a  definite  final  state  does  not  depend  upon  these 
two  states  alone,  but  also  upon  all  the  intermediate  states  through 
which  the  system  has  passed. 

These  two  examples  we  have  just  given  justify  the  following 
propositions : 

In  general,  when  a  material  system,  acted  upon  by  certain  forces, 
undergoes  a  certain  modification,  it  is  not  sufficient,  in  order  to  find 
the  work  done  by  these  forces  during  this  modification,  to  know  the 
initial  and  final  states  of  the  system;  it  is  necessary  to  know  also  the 
series  of  intermediate  states  the  system  has  passed  through  and  the 
forces  acting  upon  the  system  during  each  of  these  states. 

However,  if  the  forces  acting  upon  the  system  belong  to  certain 
particular  categories,  in  order  to  find  the  work  done  during  a  modifi- 
cation it  is  sufficient  to  know  the  initial  and  final  states;  any  knowl- 
edge of  the  intermediate  states  is  superfluous. 

10.  Potential. — Let  us  consider  this  particular  class  of  forces, 
of  which  gravity  is  an  example 

Consider  a  system  acted  upon  by  such  forces,  and  let  us  choose 
once  for  all  a  definite  state  of  the  system  which  we  shall  call  the 
state  a. 

Suppose  the  system  passes  into  this  state  a  starting  from  an- 
other state  x;  the  forces  considered  do  work  which  is  entirely 
determined  by  the  knowledge  of  the  initial  state  x  and  of  the  final 
state  a;  in  order  to  change  the  value  of  this  work,  it  would  be 
necessary  to  change  at  least  one  of  the  two  states  a  or  x;  since  we 
suppose  the  state  a  chosen  in  an  irrevocable  manner,  we  may  say 
that  the  value  of  this  work  depends  merely  on  the  choice  of  the 
state  z;  we  shall  designate  it  by  Qx.  We  propose  to  calculate  the 
work  W,  done  by  the  forces  considered,  when  the  system  passes 


WORK  AND  ENERGY.  15 

from  any  initial  condition  0  to  any  final  condition  1,  and  with  this 
object  let  us  consider  the  following  modification  : 

The  system  passes  first  from  the  state  0  to  the  state  1,  then 
from  the.  state  1  to  the  state  a;  by  definition,  the  forces  considered 
do  work,  during  the  first  part  of  the  modification,  equal  to  W, 
and  during  the  second  part  the  work  Qv  or  in  all  the  work  (W+  QJ. 

But  the  modification  considered  causes  the  system  to  pass  from 
the  state  0  to  the  state  a;  the  work  done  has  the  value  £0. 

We  have  then  the  equation 


or 

(12)  W-  £„-£,. 

When  the  work  of  the  forces  applied  to  a  system  is  entirely  deter- 
mined by  a  knowledge  of  the  initial  and  final  states,  we  may  represent 
each  state  of  the  system  by  a  quantity  Q,  variable  from  one  state  to 
another;  the  work  done  in  the  course  of  a  certain  modification  is  equal 
to  the  excess  of  the  initial  value  of  Q  over  the  -final  value  of  this  same 
quantity. 

The  quantity  Q  is  called  the  potential  of  the  forces  which  act 
upon  the  system. 

Instead  of  saying  that  the  work  of  the  forces  which  act  on  a 
system  depends  only  on  the  initial  and  final  states  of  the  system, 
we  say  that  these  forces  have  a  potential. 

11.  Potential  due  to  gravity.  —  The  comparison  of  equations 
(4)  and  (12)  show  us  that  gravity  depends  upon  a  potential  and 
that  this  potential  is 

(13)  Q  =  Mgz, 

M  being  the  total  mass  of  the  body  and  z  the  distance  of  the  centre 
of  gravity  above  an  arbitrary  horizontal  plane  H. 

12.  Forces  which  have  a  potential  in  virtue  of  the   restric- 
tions imposed  upon  the  system.  —  A  system  of  forces  which,  in 
general,  does  not  admit  a   potential  may,  in  certain  cases,  have 
one  provided  that  certain  restrictions  are  imposed  on  the  trans- 
formations to  be  studied. 

Thus,  we  have  seen  that  if  the  surface  of  a  body  supported  a 
uniform  normal  pressure,  this  pressure,  in  general,  does  not  admit 
a  potential. 


16  THERMODYNAMICS  AND  CHEMISTRY. 

But  suppose  that  we  compel  the  pressure  to  keep  an  abso- 
lutely invariable  value  P;  the  work  of  the  forces  which  act  upon 
the  system  will  be  given  by  equation  (5),  which  taken  together 
with  equation  (12)  furnishes  the  following  proposition: 

When  the  farces  which  act  on  a  system  reduce  to  a  normal,  uniform, 
constant  pressure  P,  the  forces  admit  the  potential 

(14)  Q  =  PV, 

where  V  is  the  variable  volume  of  the  system. 

Suppose,  similarly,  that  a  system  supporting  a  normal,  uni- 
form pressure  is  enclosed  in  a  reservoir  of  invariable  volume.  From 
equation  (6),  a  transformation  taking  place  under  these  condi- 
tions occasions  no  work  due  to  the  pressure.  So  that,  when  a 
system  supporting  a  normal,  uniform  pressure  keeps  an  invariable 
volume,  the  forces  which  act  upon  it  have  a  zero  potential. 

13.  Energy.  —  The  notions  of  work  and  of  potential  have  an 
extreme  importance  in  all  parts  of  mechanics,  in  statics  as  well  as 
in  dynamics;  we  shall  have  some  idea  of  this  when  treating  oiThe 
Theorem  of  Energy,  as  we  shall  do  now. 

Consider  a  system  formed  of  material  points  of  mass  m}  m't 
m",  .  .  .;  suppose  this  system  in  motion,  and  at  a  given  in- 
stant let  V,  V,  V"  .  .  .  be  the  velocities  of  the  various  points; 
multiply  the  mass  of  each  point  by  the  square  of  its  velocity;  find 
the  sum  of  the  products  thus  obtained;  finally,  take  half  of  this 
sum;  we  form  the  quantity 


This  quantity  is  called  the  kinetic  energy  of  the  system  at  the 
instant  considered. 

The  mass  of  a  material  point  is  essentially  positive;  so  is  the 
square  of  the  velocity,  unless  the  point  is  at  rest;  consequently 
the  kinetic  energy  of  a  system  is  a  quantity  essentially  positive,  unless 
the  system  is  at  rest,  in  which  case  the  kinetic  energy  is  equal  to 
zero. 

The  Theorem  of  Energy  may  be  stated  as  follows  : 

For  a  system  in  motion  during  any  lapse  of  time,  the  increase 
in  kinetic  energy,  during  this  time,  is  equal  to  the  work  done  by  the 
forces  which  act  on  the  system. 


WORK  AND  ENERGY.  17 

Suppose  that,  during  the  lapse  of  time  considered,  the  kinetic 
energy  changes  irom  the  value  WQ  to  the  value  W^,  let  W  be  the 
work  done  by  the  forces  which  act  upon  the  system;  the  preced- 
ing theorem  is  expressed  by  the  equation 

(15)  W=W,-WQ. 

14.  Principle  of  virtual  displacements. — From  this  theorem 
we  deduce  immediately  an  important  corollary. 

Suppose  that  the  initial  velocities  of  the  different  points  of  the 
system  are  a  zero;  W  will  be  equal  to  0  and  equation  (15) 
reduces  to  W=Wr  If  the  velocities  of  the  various  parts  of  the 
system  are  not  all  zero  in  the  final  state,  Wl  is  positive,  so  that  the 
following  proposition  may  be  stated: 

When  a  system,  starting  from  an  initial  state  in  which  the  veloci- 
ties of  its  various  points  are  equal  to  zero,  is  set  in  motion  and 
attains  a  state  in  which  the  velocities  of  the  various  points  are  not  all 
zero,  the  work  done  by  the  forces  which  act  on  it  is  assuredly  positive. 

When  a  system  starts  from  a  state  0  where  its  various  points 
have  zero  velocities  and  is  put  in  motion,  it  may  be  that  this 
motion  brings  it  into  anothe  state  1  where  its  variou  ?  points  still 
have  zero  velocities.  Thus,  when  a  pendulum  is  pulled  aside  a  cer- 
tain angle  to  the  left  from  its  equilibrium  position,  and  is  set  swing- 
ing without  initial  velocity,  it  returns  towards  the  equilibrium  posi- 
tion, passes  this,  and  at  the  instant  it  attains  to  the  right  an  angle 
equal  to  the  previous  devia  ion  to  the  left,  the  velocities  of  its 
various  points  all  recover  the  value  0.  But  it  is  clear  that  between 
these  two  states  in  which  the  various  points  have  their  velocities 
equal  to  zero,  the  system  passes  through  a  continuous  series  of 
states  in  which  certain  of  the  points  have  velocities  different  from 
0;  in  other  words,  a  system  cannot  quit  a  state  in  which  the  veloci- 
ties of  its  various  points  are  zero  except  by  first  traversing  a  series 
of  states  in  which  these  velocities  are  not  all  zero. 

From  the  preceding  theorem,  taking  account  of  the  last  remark, 
the  following  proposition  is  evident: 

A  system  placed  without  initial  velocity  in  a  given  state  cannot  be 
put  in  motion  unless  the  beginning  of  this  motion  corresponds  to 
positive  work  of  the  forces  applied  to  the  system. 

This  proposition  will  furnish  us  a  means  of  recognizing  if  a 


18  THERMODYNAMICS  AND  CHEMISTRY. 

material  system,  acted  upon  by  given  forces,  is  surely  in  equilib- 
rium in  a  given  state. 

Let  us  consider  the  various  ways  in  which  the  system  may  be 
removed  from  the  given  state  and  the  different  displacements  from 
this  state  it  may  be  supposed  to  take;  each  of  these  imagined  dis- 
placements is  called  a  virtual  displacement. 

If  the  commencement  of  every  virtual  displacement  imposed  on 
a  system  in  a  given  state  corresponds  to  negative  or  zero  work  of  the 
forces  applied  to  the  system,  the  latter,  placed  without  initial  velocity 
in  the  state  considered,  will  remain  in  it  in  equilibrium. 

This  proposition  has  the  name  of  the  Principle  of  Virtual  Dis- 
placements. 

15.  Conservation  of  energy.  Conservative  systems.— The 
various  propositions  we  have  stated  assume  a  remarkable  and 
simple  form  when  the  system  studied  is  acted  upon  by  forces  which 
admit  a  potential  Q.  In  this  case,  the  work  done  by  these  forces 
is  expressed  by  means  of  equations  (12)  and  (15),  and  the  theorem 
of  energy  becomes 

(16)  W,-W.=Q»-Qr 

When  a  system  acted  upon  by  forces  which  depend  on  a  potential 
is  in  motion,  the  increase  in  kinetic  energy  during  a  certain  time  is 
equal  to  the  decrease  in  the  potential  during  the  same  time. 

Equation  (16)  may  also  be  written 

(160  Bi  +  W^Q^W,, 

and  the  preceding  theorem  be  stated  as  follows: 

When  a  system  is  in  motion  under  the  action  of  forces  which  admit 
a  potential,  the  sum  of  the  potential  and  the  kinetic  energies  maintains 
an  invariable  value  throughout  the  duration  of  the  motion. 

Consider  a  system  having  a  motion  which,  after  a  certain  time, 
brings  it  back  to  its  initial  state,  such  as  a  pendulum  after  a  com- 
plete oscillation;  the  potential  will  retake  its  initial  value;  accord- 
ing to  the  preceding  proposition,  the  same  will  be  true  of  the  kinetic 
energy;  from  this,  the  name  principle  of  the  conservation  of  energy  is 
given  to  the  preceding  proposition,  and  the  name  conservative 
systems  is  sometimes  given  to  systems  acted  upon  by  forces  that 
depend  on  a  potential. 


WORK  AND  ENERGY.  19 

To  illustrate,  let  us  apply  equation  (16')  to  a  system  formed 
of  a  single  material  point  of  mass  m.  By  equation  (13)  we  shall 
have  Q  =  mgz,  z  being  the  distance  of  the  point  above  an  arbitrary 
horizontal  plane,  while  the  kinetic  energy  reduces  to  W=$mV2. 
Noting  that  the  mass  m  of  the  material  point  is  a  quantity  of  in- 
variable magnitude,  we  see  that  equation  (160  becomes 


and  includes  the  following  proposition: 

Every  time  that  a  material  particle  passes  through  a  given 
level  it  has  a  velocity  whose  direction  may  have  changed,  but 
whose  value  is  ever  the  same. 

16.  Principle  of  virtual  displacements  for  conservative  sys- 
tems. Stability  of  equilibrium.  —  In  order  that  the  forces  which 
act  upon  a  conservative  system  do  positive  work,  it  is  necessary 
and  sufficient  that  the  potential  of  the  system  decreases.  From 
this,  starting  from  a  certain  state  in  which  the  velocities  of  its 
various  points  are  all  zero,  a  conservative  system  cannot  attain 
another  state  in  which  some  at  least  of  these  velocities  are  differ- 
ent from  0,  unless  the  potential  has  a  less  value  in  the  second  state 
than  in  the  first. 

In  particular,  a  conservative  system  cannot  quit  a  state  in 
which  its  various  points  have  zero  velocities,  unless  the  potential 
decreases,  at  least  at  the  beginning  of  the  motion.  If  then  all  the 
virtual  displacements  that  may  be  imposed  upon  a  conservative 
system  in  a  given  state  commence  by  causing  the  potential  to 
increase  or  to  keep  its  value  constant,  the  system  is  assuredly  in 
equilibrium. 

All  the  virtual  displacements  imposed  on  a  system  in  a  given 
state  will  commence  necessarily  by  increasing  the  potential  if, 
in  this  state,  the  potential  has  a  smaller  value  than  in  all  neigh- 
boring states;  whence  the  following  theorem: 

A  conservative  system  is  surely  in  equilibrium  if  placed,  without 
initial  velocity,  in  a  state  in  which  the  potential  has  a  minimum  value. 

It  is  shown  in  mechanics  by  methods  that  we  cannot  expose 
here  that  the  following  proposition  is  true: 

A  conservative  system  is  surely  in  STABLE  equilibrium  when  it  is, 


20  THERMODYNAMICS  AND  CHEMISTRY. 

without  initial  velocity,  in  a  state  in  which  the  potential  has  a  minimum 
value. 

Let  us  apply  these  last  propositions  to  a  system  having  weight. 

The  potential  of  such  a  system  is  given  by  the  equation 

(13)  Q  =  Mgz, 

in  which  M  is  the  mass  of  the  system  and  z  the  distance  of  the 
centre  of  gravity  above  an  arbitrary  horizontal  plane;  the  weight 
Mg  of  the  system  being  a  positive  quantity  of  invariable  value, 
the  preceding  theorems  give  us  the  following  propositions,  which 
have  an  important  place  in  the  development  of  mechanics: 

A  system  having  weight,  placed  without  initial  velocity  in  a  certain 
state,  cannot  quit  it  without  occasioning  a  lowering  of  its  centre  of 
gravity. 

A  system  having  weight  is  surely  in  stable  equilibrium  when  its 
centre  of  gravity  is  as  low  as  possible. 


CHAPTER  II. 
QUANTITY   OF   HEAT  AND   INTERNAL   ENERGY. 

17.  Loss  of  kinetic  energy  and  generation  of  heat  in  a  machine 
left  to  itself. — Consider  what  in  the  older  mechanics  was  called  a 
machine,  that  is,  a  more  or  less  complicated  arrangement  of  bodies 
each  of  which  is  movable,  but  keeping  invariable  its  size,  form, 
and  state. 

Put  this  machine  in  motion;  its  various  parts  having  velocities 
different  from  0,  its  kinetic  energy  will  also  have  a  value  W0  that 
is  positive. 

Abandon  this  machine  to  itself  or,  to  speak  more  precisely, 
let  the  system  move  so  that  no  force  acts  upon  it;  will  the  machine 
continue  indefinitely  in  motion? 

If,  to  reply  to  this  question,  we  consult  the  theorems  announced 
in  the  preceding  chapter,  we  shall  be  led  to  conclude  that  the 
motion  of  our  machine  will  be  indefinitely  conserved;  thus,  the 
forces  which  act  upon  the  system,  being  always  zero,  will  do  no 
work;  the  kinetic  energy  of  the  system  will  keep  forever  an  in- 
variable value;  positive  at  the  start,  it  can  never  become  zero  and, 
set  going,  our  machine  will  always  have  some  part  in  motion. 

Now,  this  reply  is  evidently  contrary  to  the  experience  that 
every  machine  teaches  us:  left  to  itself,  it  stops  sooner  or  later. 
The  kinetic  energy  of  the  system,  far  from  remaining  constant, 
decreases  incessantly.  To  keep  the  machine  in  motion,  to  keep 
the  kinetic  energy  constant,  it  is  necessary  to  rubmit  this 
machine  to  the  action  of  forces  which  do  positive  work  upon  it 
unceasingly. 

We  see  from  these  observations  that  real  machines  differ  notably 
from  ideal  machines  to  which  the  theorems  of  the  first  chapter 
apply.  Other  differences  may  also  be  made  evident;  these  ideal 

21 


22  THERMODYNAMICS  AND  CHEMISTRY. 

machines  are  made  of  bodies  whose  state  is  invariable;  on  the  con- 
trary, the  bodies  of  which  a  real  machine  is  composed  undergo  a 
modification:  they  heat  up,  their  temperature  increases;  to  keep 
them  in  an  invariable  state,  to  keep  their  temperature  constant, 
it  is  necessary  to  cool  them,  to  oblige  them  to  cede  a  certain  quan- 
tity of  heat  to  foreign  bodies. 

We  are  thus  led  to  recognize,  for  every  system  in  motion  which 
is  formed  of  bodies  kept  in  an  invariable  state  and  temperature, 
the  two  following  properties,  the  first  of  which  contradicts  the 
theorem  of  energy,  and  the  second  cannot  be  predicted  by  the 
principles  set  forth  in  the  first  chapter: 

1°.  The  work  W  done  by  the  forces  which  act  upon  the  system 
during  a  definite  lapse  of  time  surpasses  the  increase  (Wl  —  W0) 
which  the  kinetic  energy  undergoes  during  the  same  time: 

W-(Wl-W0)>0. 

2°.  The  quantity  of  heat  liberated  by  the  system  during  the 
same  lapse  of  time  is  positive: 


18.  Mechanical  equivalent  of  heat.  —  A  natural  and  simple 
hypothesis  consists  in  supposing  that  the  two  quantities 
(W—  W^  +  WQ)  and  Q,  whose  sign  is  always  the  same,  are  in  a 
constant  ratio,  or,  in  other  words,  to  suppose  that  there  exists  a 
definite  positive  number  E,  such  that 


This  number  E  is  called  the  mechanical  equivalent  of  heat. 
The  correctness  of  the  preceding  proposition  is  subordinate  to  the 
following  conditions:  the  bodies  which  constitute  the  system  are 
movable,  but  the  size,  form,  state,  and  temperature  of  each  re- 
mains invariable;  W  represents  the  work  of  all  the  forces  that  act 
upon  the  system,  including  the  forces  that  originate  in  bodies  ex- 
terior to  the  system  as  well  as  forces  by  which  the  various  parts  of 
the  system  act  upon  each  other. 

19.  Principle  of  the  equivalence  of  heat  and  work.  —  The 
usefulness  of  the  preceding  principle  becomes  greatly  limited  from 


QUANTITY  OF  HEAT  AND  INTERNAL  ENERGY.          23 

the  obligations  to  consider  only  collections  of  bodies  each  of  which 
keeps  an  invariable  size,  form,  temperature,  and  state;  so  that  it 
.s  advantageous  to  leplace  this  principle  by  another  which  is 
analogous  but  not  ident.cal. 

To  sta.e  this  new  principle,  we  will  make  use  of  an  idea  that 
we  shall  meet  often  in  what  follows,  the  idea  of  closed  cycle. 

When,  at  the  close  of  a  transformation,  the  bodies  which  com- 
pose a  system  reassume  the  arrangement,  size,  form,  temperature, 
physical  and  chemical  states  that  they  possessed  at  the  beginning 
of  the  transformation,  we  say  that  by  the  effect  of  this  trans- 
formation the  system  has  described  a  closed  cycle. 

The  various  parts  of  the  system  cannot  possess  the  same  veloci- 
ties at  the  beginning  and  at  the  end  of  a  closed  cycle,  so  that  the 
kinetic  energy  cannot  retake  its  initial  value  at  the  moment  the 
system  finishes  describing  a  closed  cycle. 

Suppose  that  a  system  d  scribes  a  closed  cycle  during  which  there 
is  set  free  a  quantity  of  heat  Q  while  the  kinetic  energy  passes  from 
the  value  W0  to  the  value  W^,  let  We  denote  the  EXTERNAL  WORK, 
that  is,  the  work  done  by  the  forces  that  bodies  foreign  to  the  system 
exert  on  this  system;  we  shall  then  have  the  equation 

(2)  E*= 


This  equation  may  also  be  written 
(2r)  W^Wi  +  W^EQ. 

This  fundamental  hypothesis  is  what  we  shall  call  the  Principle 
of  he  equivalence  of  heat  and  mechanical  work. 

20.  Value  of  the  mechanical  equivalent  of  heat.  —  Depending 
on  this  principle  and  on  the  results  of  various  experiments 
whose  description  is  beyond  the  scope  of  this  work,  Robert  Mayer, 
Joule,  and  a  great  number  of  other  physicists  have  occupied  them- 
selves with  the  determination  of  the  value  of  the  mechanical  equiv- 
alent of  heat  E'f  they  found  that  approximately 

#=425, 

when  the  gramme-metre  is  taken  as  unit  of  work  and  the  gramme- 
calorie  or  small  calorie  as  unit  of  heat. 


24  THERMODYNAMICS  AND   CHEMISTRY. 

If,  instead  of  the  gramme-metre,  the  unit  of  work  taken  is  the 
erg,  which  is  98100  times  smaller,  the  numerator  of  equation  (2) 
is  represented  by  a  number  98100  times  greater;  the  value  of  the 
denominator  does  not  change  if  the  small  calorie  is  kept  as  the 
heat-unit,  the  new  value  of  the  mechanical  equivalent  is  then 

#=425X98100  =  41692500. 

21.  Extension  of  the  principle  of  the  equivalence  of  heat  and 
work  to  an  unclosed  cycle.  —  The  principle  of  the  equivalence 
between  heat  and  work,  such  as  we  have  stated  it  in  Art.  19,  re- 
quires that  the  modification  to  which  it  is  desired  to  apply  it 
is  a  closed  cycle;  this  restriction  is  sometimes  embarrassing;  we 
shall  obviate  this  by  modifying  the  statement  of  the  principle. 

To  shorten  our  terms,  let  us  denote  by  e  the  quantity 


The  principle  of  equivalence  may  be  thus  announced  : 
If  a  system  describes  a  closed  cycle,  the  quantity  e,  calculated 
for  the  whole  of  this  closed  cycle,  is  equal  to  zero. 

Keeping  this  in  mind,  imagine  a  system  to  pass  from  a  given 
initial  state  0  to  a  final  state  1,  by  a  definite  series  of  modifications; 
the  quantity  e,  calculated  for  this  first  transformation,  takes  a 
certain  value  that  we  may  call  e.  Suppose  now  that  the  system 
returns  from  the  state  1  to  the  state  0  by  an  equally  definite  series 
of  modifications;  the  quantity  e  has,  fo  this  second  transforma- 
tion, a  certain  value  )?;  for  the  total  transformation,  the  quantity 
e  has  the  value  (e  +  ^),  but  ince  this  total  transformation  is  a 
closed  cycle,  we  have 

£+)?=0. 

Suppose  now  that  the  system  passes  from  the  same  initial 
state  0  to  the  same  final  state  1,  but  by  a  different  series  of  modi- 
fications than  in  the  preceding  case;  for  this  transformation  the 
quantity  e  will  be  equal  to  e';  then  bring  it  back  from  state  1  to 
state  0  by  the  same  transformation  as  in  the  preceding  case,  for 
which  the  value  of  e  is  y;  this  time  we  shall  have 


QUANTITY  OF  HEAT  AND  INTERNAL  ENERGY.         25 
The  two  equalities  require  that 


£=£' 


so  that  the  principle  of  the  equivalence  between  heat  and 
work  leads  to  this  consequence:  in  whatever  manner  a  system 
passes  from  a  given  initial  state  to  a  definite  final  state,  the  quan- 
tity e  keeps  the  same  value. 

If  is  clear,  also,  that  this  proposition  includes  also  the 
principle  of  the  equivalence  between  heat  and  work;  thus,  if  the 
system  describes  a  closed  cycle,  e  will  have,  for  this  cycle,  the  same 
value  as  for  any  other  transformation  that  takes  the  system  in 
the  same  initial  state  and  returns  it  to  this  state;  or  the  same 
value  as  if  the  system  were  not  modified  at  all,  and  this  last  value 
is  evidently  0;  so  that,  for  any  closed  cycle,  e=0;  this  is  precisely 
the  original  statement  of  the  principle  of  the  equivalence  between 
heat  and  work. 

We  are  therefore  authorized  to  state  in  the  following  way  the 

PRINCIPLE  OF  THE  EQUIVALENCE  BETWEEN  HEAT  AND  WORK  I 

The  quantity 


has  a  value  which  depends  on  the  initial  and  final  states  of  the  system, 
but  not  on  the  nature  of  the  modifications  undergone  by  the  system  in 
order  to  pass  from  this  initial  state  to  this  final  state. 

22.  Internal  energy.  —  It  is  now  sufficient  to  follow  a  reasoning 
similar  to  that  which  gave  us  the  idea  of  potential  (Art.  10)  to 
transform  the  preceding  statement  into  the  following: 

For  each  state  X  of  the  system  there  may  be  made  to  correspond  a 
quantity  Ux  such  that,  for  every  modification  undergone  by  the  system 
from  the  initial  state  0  to  the  final  state  1,  this  equality  holds: 

e=U  -U^, 

or  again,  replacing  e  by  its  value,  defined  by  equation  (3), 
(4)  EQ-We=W0- 


Such  is  the  form  given  in  1850,  by  R.  Clausius,  to  the  principle 
of  the  equivalence  between  heat  and  work. 


26  THERMODYNAMICS  AND  CHEMISTRY. 

To  the  quantity  Ux  Clausius  gave  the  name  Internal  Energy 
of  the  system  in  the  state  X. 

The  preceding  equalities  show  us  that  Ux  is  a  quantity  of  the 
same  kind  as  e  and  Q,  that  is,  it  may  be  measured  in  heat-units. 

Certain  writers,  instead  of  keeping  a  special  name  for  the 
quantity  Ux,  prefer  to  consider  the  product  EUX,  which  they  call 
the  potential  energy  of  the  system  in  the  state  X;  the  potential 
energy  is  then  a  quantity  of  the  same  kind  as  We  and  Wl}  or  a 
quantity  measured  in  units  of  work;  we  may  say  that  the  potential 
energy  is  the  equivalent  oj  the  internal  energy.  The  quantity 

EUX  +  WX 

may  be  called  the  total  energy  of  the  system  in  the  state  X. 

23.  Principle  of  the  conservation  of  energy.—  That  an  in- 
terest attaches  itself  to  this  nomenclature  is  seen  from  the  follow- 
ing: 

Imagine  a  system  isolated  in  space  and  consequently  deprived 
of  the  action  of  any  external  body. 

No  external  body  exerting  force  upon  this  system,  every  modi- 
fication that  it  undergoes  is  accompanied  by  external  work 
equal  to  0: 

We=0. 

No  external  body  can  give  to  or  take  heat  from  this  system; 
whatever  modification  it  undergoes,  the  quantity  of  heat  set  free 
can  be  neither  positive  nor  negative;  it  is  zero: 


Thus,  for  such  a  system,  equation  (4)  becomes 


and  is  stated: 

Whatever  modification  an  isolated  system  undergoes,  this  modi- 
fication leaves  the  total  energy  of  the  system  unchanged;  what  of  the 
potential  energy  is  lost  is  gained  in  kinetic  energy,  and  conversely. 

This  proposition  bears  the  name  of  the  principle  of  the  con- 
servation of  energy.  It  is  quite  unnecessary  to  assign  to  it  a  vague 
metaphysic  sense  or  a  mysterious  origin;  it  is  simply  a  special 
case  of  a  physical  law,  the  law  of  the  equivalence  of  heat  and  work. 


QUANTITY  OF  HEAT  AND  INTERNAL  ENERGY.          27 

24.  Gases  which  obey  Mariotte's  Law.  Absolute  tempera- 
ture.— Before  applying  the  principle  of  the  equivalence  of  heat 
and  work  to  the  various  problems  of  chemical  calorimetry,  we 
shall  make  an  application  which  we  shall  employ  in  what  follows. 

All  are  acquainted  with  the  statement  of  Mariotte's  Law:  at 
a  given  temperature,  the  product  of  the  pressure  supported  by  a 
mass  of  gas  and  the  volume  it  occupies  is  a  constant.  Every- 
body knows  also  that  this  law  does  not  apply  rigorously  to  any 
gas,  but  that  the  gases  distant  from  the  conditions  in  which  they 
liquify  obey  this  law  approximately. 

It  results  from  this  law,  as  is  taught  in  elementary  classes,  that 
the  two  coefficients  of  expansion,  under  constant  pressure  and  con- 
stant volume,  of  a  gas  obeying  this  law  have  the  same  value,  and 
this  with  whatever  thermometer  used;  besides,  according  to  the 
observations  of  Charles  and  of  Gay-Lussac,  this  value  is  the  same 
for  all  gases  which  follow  sensibly  Mariotte's  Law;  finally,  if  the 
thermometer  chosen  is  one  constructed  with  one  of  these  gases, 
the  value  in  question  evidently  does  not  depend  upon  the  tem- 
perature. 

If  the  temperature  chosen  is  a  centigrade  temperature,  the 
coefficient  of  expansion  of  gases  which  obey  Mariotte's  Law  is  sensi- 
bly, according  to  Regnault's  observations,  a=  ^-y. 

At  two  different  temperatures  t  and  t',  and  under  two  different 
pressures  P  and  P',  the  same  mass  of  gas  will  occupy  the  volumes 
V  and  F',  joined  by  the  relation 


P'V 

<6>  r£r— 


This  relation  may  also  be  written 
PV     P'V 


a.          a 

The  expressions  T=-+t  and  T'=-+t',  which  appear  in  this 

equation,  are  called,  for  reasons  which  we  cannot  go  into  here,  the 
absolute  temperatures  corresponding  to  the  centigrade  temperatures 


28 


THERMODYNAMICS  AND  CHEMISTRY. 


t  and  t'.    The  absolute  temperature  which  corresponds  to  a  given 
centigrade  temperature  is  thus  obtained  by  adding  to  the  latter 

the  number  -,  equal  to  about  273.    The  difference  between  two 

absolute  temperatures  is  equal,  by  definition,  to    the   difference 
between  the  two  corresponding  centigrade  temperatures: 

(7)  T'-T=t'-t. 

Making  use  of  absolute    temperatures,  equation  (6)  may  be 
written 

PV    P'V' 


25.  Expansion  of  a  gas  in  vacuo.     Gay-Lussac's  experiment. 

—The  gases  which  obey  Mario  tte's  Law  approximately,  obey  also, 

approximately,  another  law  illustrated  by  an  old  experiment  of 

Gay-Lussac,  repeated  by  Regnault,  by  W.  Thomson,  and  by  Joule. 

Two  reservoirs  R,  R'  (Fig.  17),  one  of  volume  v,  the  other  of 


FIG.  17. 

volume  v',  are  immersed  in  a  calorimeter;  a  tube  fitted  with  a 
stop-cock  r  joins  the  two  reservoirs. 

At  the  beginning  of  the  experiment,  the  cock  r  is  closed,  the 
reservoir  R  is  full  of  gas  and  the  reservoir  R'  is  empty;  the  water 
in  the  calorimeter  has  a  certain  temperature. 

When  the  cock  r  is  opened,  a  part  of  the  gas  that  filled  the 
reservoir  R  passes  into  the  reservoir  Rf. 


QUANTITY  OF  HEAT  AND  INTERNAL  ENERGY.         29 

At  the  end  of  the  experiment  the  two  reservoirs  R  and  Rf  are 
filled  with  gas  at  the  same  density;  it  is  observed  that  the  water 
in  the  calorimeter  has  the  same  temperature  as  at  the  beginning. 

Let  us  apply  to  this  transformation  the  principle  of  the  equiva- 
lence of  heat  and  work,  expressed  by  equation  (4). 

At  the  beginning,  as  at  the  end  of  the  transformation,  the  gas 
is  at  rest;  the  reservoirs  do  not  move;  the  kinetic  energy  is  zero  : 


The  exterior  forces  applied  to  this  system,  the  pressures  exerted 
on  the  external  surface  of  the  reservoirs,  have  done  no  work,  for 
this  surface  has  not  moved: 


Finally,  since  the  water  in  the  calorimeter  returns  to  its  initial 
temperature,  the  system  considered  has  neither  given  out  nor 
absorbed  heat: 

<2=0. 

Then  equation  (4)  gives  us 


The  final  value  of  the  internal  energy  of  the  system  is  equal 
to  its  initial  value. 

The  system  studied  consists  of  two  parts:  the  matter  which 
forms  the  walls  of  the  reservoirs,  and  the  gas  enclosed  by  these 
reservoirs;  the  matter  which  forms  the  walls  of  the  reservoirs  has 
undergone  no  modification,  so  that  its  internal  energy  has  not 
changed  in  value;  thus  the  internal  energy  of  the  ga's  should  have, 
at  the  end  of  the  transformation  considered,  the  same  value  as 
at  the  start.  Nevertheless,  this  gas  has  undergone  a  change:  its 
temperature,  it  is  true  is  the  same  at  the  'beginning  and  at  the 
end  of  the  transformation;  but  its  volume,  which  was  v,  has 
become  (v  +v');  we  are  thus  led  to  announce  the  following  law: 

//  he  volume  of  a  mass  of  gas  is  varied,  while  its  temperature  is 
kept  constant,  the  internal  energy  of  this  body  undergoes  no  change 
in  value. 


30  THERMODYNAMICS  AND  CHEMISTRY. 

26.  Perfect  gases. — This  law  has  the  same  limitations  as  Ma- 
rio tte's  Law;  no  gas  obeys  it  rigorously,  but  certain  gases  taken  in 
proper  conditions  obey  it  very  closely,  and  these  gases  are  precisely 
those  that  follow  Mario tte's  Law  most  exactly;  such  are,  under 
the  ordinary  conditions  of  temperature  and  pressure,  hydrogen, 
oxygen,  nitrogen,  carbonic  oxide,  nitrous  oxide;   we  give  to  these 
gases  the  name  of  gases  near  to  the  ideal  state,  keeping  the  name 
perfect  gas  for  an  ideal  gas  which  would  obey  exactly  the  two  laws 
just  cited. 

Although  a  perfect  gas  does  not  exist  any  more  than  does  an 
absolutely  rigid  body,  the  study  of  ideal  gases  in  thermodynamics 
is  as  legitimate  and  as  useful  as  the  study  of  rigid  bodies  in  me- 
chanics; it  furnishes  an  image  simplified  and  approximate,  it  is 
true,  but,  in  a  great  number  of  cases,  practically  sufficient,  of  the 
properties  of  real  gases. 

27.  Specific  heat  at  constant  volume.     Internal  energy  of  a 
perfect  gas. — Bring  a  gas  from  temperature  tQ  to  temperature  t, 
keeping  it  in  a  reservoir  of  volume  V.     Let  M  be  the  mass  of  this 
gas  expressed  in  grammes.     It  absorbs,  during  the  transformation 
considered,  a  quantity  of  heat  whose  value  q  is  expressed  in  gramme- 
calories  or  in  small  calories;  other  things  being  equal,  q  is  propor- 
tional to  M . 

The  ratio  ,,,          =c  is,  by  definition,  the  mean  value,  between 

M.  (t  LQ) 

the  temperatures  t0  and  t,  of  the  specific  heat  at  constant  volume  of 
the  gas  con  idered. 

Let  us  apply  equation  (4)  to  the  transformation  considered. 

The  volume  of  the  gas  remains  constant;  then,  by  formula  (6) 
of  the  preceding  chapter,  the  work  done  by  the  pressure  the  foreign 
bodies  exert  on  the  gas  is  equal  to  0 : 

TF,=0. 

The  gas  is  at  rest  at  the  beginning  and  at  the  end  of  the  trans- 
formation, so  that  TF0=0,  W=Q. 

The  quantity  Q  of  heat  set  free  is  equal  to  —q,  so  that 

Q=-Mc(t-t0). 


QUANTITY  OF  HEAT  AND  INTERNAL  ENERGY.          31 

If  then  U0  is  the  internal  energy  of  the  mass  of  gas  at  volume 
V  and  temperature  tQ,  and  if  U  is  the  internal  energy  of  this  same 
mass  at  the  same  volume  and  at  temperature  t,  equation  (4)  be- 
comes 

(8)  U-U0=Mc(t-t0). 

The  same  gas  might  be  heated  between  the  same  limits  of 
temperature,  but  keeping  it  at  a  volume  V  different  from  V. 
From  what  we  have  seen  in  the  preceding  article,  if  the  gas  is  a 
perfect  one,  the  values  of  U0  and  U  will  undergo  no  change;  and 
similarly  for  c.  Thus  the  mean  value  between  two  given  tempera- 
tures of  the  specific  heat  at  constant  volume  of  a  perfect  gas  does  not 
depend  upon  the  value  of  this  volume  at  which  the  gas  is  kept. 

If  we  suppose  the  temperatures  t0  and  t  read  on  a  thermometer 
constructed  of  a  sensibly  perfect  gas,  we  shall  have,  by  equation  (7) 

t-t0=T-TQ, 
and  equation  (8)  becomes 

(9)  U=U0-McT0+McT. 

28.  Specific  heat  at  constant  pressure.  Robert  Mayer's 
Relation.  —  Let  us  take  now  the  same  mass  M  of  the  same  gas, 
always  expressed  in  grammes;  in  equilibrium  at  the  temperature 
t0,  under  a  certain  pressure  P,  it  occupies  a  certain  volume  F0;  in 
equilibrium  at  the  temperature  t,  and  under  the  same  pressure  P, 
it  occupies  a  volume  V.  By  heating  in  this  way  at  constant 
pressure,  from  the  temperature  t0  to  the  temperature  t,  it  absorbs 

a  certain  quantity  of  heat  q'.     By  definition,  the  ratio 


=  C  is  the  mean  value,  between  the  temperatures  tQ  and  t,  of  the 
specific  heat  at  constant  pressure  of  the  gas  considered. 

Let  us  apply  equation  (4)  to  the  transformation  considered. 

The  kinetic  energy  of  the  gas  is  zero  at  the  beginning  and  at 
the  end  of  the  transformation: 

TF0=0,        W=0. 


32  THERMODYNAMICS  AND   CHEMISTRY. 

The  pressure  being  kept  constant,  the  external  work  is  given 
by  equation  (5)  of  the  preceding  chapter: 

We=P(VQ-V). 

The  quantity  of  heat  set  free  has  the  value  Q=  —  q',  or 
Q=-MC(t-t0). 

If,  then,  we  denote  by  U0  the  initial  value  and  by  U  the  final 
value  of  the  internal  energy  of  the  gas,  equation  (4)  becomes 

U-U0=MC(t-t0)-?(V-V0). 

This  equality  is  general. 

Suppose  now  that  we  have  to  deal  with  a  gas  near  to  the  per- 
fect state;  the  value  of  the  internal  energy  at  a  given  tempera- 
ture is  constant;  if  then  the  temperatures  t0  and  t  are  the  same  in 
the  preceding  equation  as  in  equation  (8),  the  values  UQ  and  U  will 
be  also  the  same  in  these  two  equalities;  equating  them  gives 


A  very  simple  transformation  will  allow  us  to  put  this  equation 
into  a  celebrated  form. 

Suppose  the  temperature  read  upon  a  thermometer  of  a  sensi- 
bly perfect  gas;  we  have  t—t0=T—TQ,  T  and  TQ  being  the  abso- 
lute temperatures  which  correspond  to  the  centigrade  temperatures 
t  and  tj  from  this  the  preceding  equation  becomes 

(10)  M(C-c)(T-T0)  =  ?(V-Vj. 

'Furthe-,  let 

6  be  the  absolute  temperature  of  melting  ice,  about  273°; 
Trthe  normal  atmospheric  pressure  in  the  system  of  units 

chosen  ; 
o  the  volume  occupied  by  one  gramme  of  the  gas  studied,  at 

the  temperature  of  melting  ice  and  under  normal  atmos- 

pheric pressure. 


QUANTITY  OF  HEAT  AND  INTERNAL  ENERGY.          33 

In  the  same  conditions,  M  grammes  of  this  gas  occupy  a  volume 
Mo,  such  that  the  equation  (6')  may  be  written 


T    ~  T0~     6    ' 

These  equations  in  their  turn  may  be  written  somewhat  differ- 
ently. 

Whatever  be  the  nature  and  mass  of  the  gas  considered,  the 
quotient 

(12)  *=R 

has  the  same  value,  which  depends  solely  on  the  mechanical  units 
employed;  thus  in  the  system  hi  which  the  unit  of  length  is  the 
metre  and  the  unit  of  force  the  gramme-weight,  we  have 

10333000 


In  the  C.G.S.  system,  in  which  the  unit  of  length  is  the  centi- 
metre and  the  unit  of  force  the  dyne,  we  have 

1033.3X981 
R=     ~273~       r3713- 

By  means  of  equation  (12)  the  equations  (11)  may  be  written 

(13)  PV=MRoT.        PV0=MRoT0, 
and  equation  (10)  becomes  simply 

(14)  C-c-f. 

This  is  ROBERT  MAYER'S  RELATION,  given  by  the  illustrious 
Hielbronn  physician  at  the  dawn  of  thermodynamics,  and  of 
which  we  shall  see  many  applications. 

29.  Influence  of  temperature  on  the-  specific  heats  of  perfect 
gases.  Clausius*  Law.  —  According  to  the  definition  which  has 
been  given  of  the  two  quantities  c  and  C,  the  value  of  each  of  them 
may  very  well  depend  upon  the  two  extreme  temperatures  t  and  £0; 
besides,  we  know  that  the  value  of  c  does  not  depend  upon  the 
constant  volume  at  which  the  gas  is  heated;  but  we  are  ignorant 
if  the  value  of  C  depends  on  the  constant  pressure  supported  by 


34  THERMODYNAMICS  AND  CHEMISTRY. 

the  gas  while  it  is  carried  from  the  temperature  t0  to  the  tempera- 
ture t. 

Mayer's  relation  teaches  us  that  the  difference  (C— c)  has,  for 
a  given  gas,  an  absolutely  definite  value;  it  follows  then,  since  for 
a  given  gas  the  value  of  c  can  depend  only  upon  the  temperatures 
t0  and  t,  that  it  is  the  same  with  C;  whence  this  first  proposition: 
The  specific  heat  at  constant  pressure  of  a  given  perfect  gas  does  not 
depend  upon  the  value  of  the  constant  pressure  under  which  the  gas 
is  heated. 

Besides,  if  we  determine  in  what  manner  one  of  the  two  quan- 
tities C,  c,  depends  on  the  two  temperatures  t0,  t,  Mayer's  relation 
will  show  us  immediately  how  the  other  depends  on  these  same 
temperatures. 

Regnault  measured,  under  constant  atmospheric  pressure,  the 
mean  specific  heats  of  various  gases  at  different  temperatures. 

For  air  the  following  numbers  were  found : 

Between  Z0=  -  30°  C.  and  t=  +  10°  C.,  0=0.23771 
0°  100°  0.23741 

0°  200°  0.23751 

More  recently  Witkovski  has  obtained  for  the  same  gas  at  the 
same  pressure: 

Between  t0=  +  20°  C.  and  t=  +  98°  C.,  C=0.2372 

-  77°  +16°  0.2374 

-102°  +17°  0.2372 

-170°  +18°  0.2427 

According  to  Regnault,  the  mean  specific  heat  of  hydrogen, 
under  atmospheric  pressure,  has  the  same  value  between  0°  and 
200°  as  between  -30°  and  +110°. 

These  observations,  joined  to  Mayer's  relation,  justify  the 
following  law,  called  CLAUSIUS'  LAW. 

The  specific  heat  at  constant  pressure  and  the  specific  heat  at  con- 
stant volume,  for  a  given  gas  near  to  the  perfect  state,  have  fixed 
values. 

The  truth  of  this  law  for  very  high  temperatures  has  been 
questioned,  notably  by  Mallard  and  Le  Chatelier;  but  their  very 
complex  experiments  can  be  interpreted  only  by  means  of  a 


QUANTITY  OF  HEAT  AND  INTERNAL  ENERGY.         35 

certain  number  of  hypotheses,  some  of  which  are  in  contradiction 
with  known  facts;  thus  these  authors  suppose  that  carbonic  acid 
gas  is  undecomposable  by  heat  up  to  1800°  and  water  vapor  up  to 
2300°,  which  is  contrary  to  the  direct  observations  of  H  Sainte- 
Claire  Deville.  We  think  then  that  Clausius'  law  may  be  con- 
served until  further  notice,  even  for  very  high  temperatures. 

30.  Evaluation   of  the  mechanical  equivalent  of  heat. — It 
is  evident  that  Mayer's  equation  (14)  may  also  be  written 


C 

Ra     c 


(140 


All  the  quantities  in  the  second  member  are  accessible  to  ex- 
periment. 

We  have  seen  how  R  may  be  calculated. 

We  have  mentioned  Regnault's  experiments  which  gave  C. 

If  u  is  the  volume  occupied  by  1  gramme  of  air  under  normal 
conditions  of  temperature  and  pressure,  if  A  is  the  density  of  the 

gas  considered  with  respect  to  air,  we  have  <T=-T,  so  that  a  may 

be  determined. 

Finally,  under  the  pressure  P,  at  the  temperature  t,  sound  is 
propagated  in  the  gas  considered  with  a  velocity  V,  which,  by 
Laplace's  formula,  has  the  value 


T=273+t  being  the  absolute  temperature  which  corresponds  to 
the  centigrade  temperature  t.    The  experimental  determination 

of  the  velocity  V  allows  the  computation  of  — . 

We  see  then  that  equation  (14')  gives  a  means  of  calculating 
the  mechanical  equivalent  of  heat;  it  is  the  method  which  led 
Robert  Mayer  to  the  first  evaluation  ever  published  of  this  quan- 
tity; before  Mayer,  Sadi  Carnot  had  obtained  a  value  of  the  me- 
chanical equivalent,  probably  by  the  same  method. 


CHAPTER  III. 
CHEMICAL  CALORIMETRY. 

31.  The  quantity  of  heat  set  free  by  a  system  which  under- 
goes a  transformation  does  not  depend  solely  upon  the  initial 
and  final  states. — The  chemists  who  occupied  themselves  with 
calorimetry,  from  the  time  of  Lavoisier  and  Laplace  down  to  the 
time  of  the  foundation  of  thermodynamics,  have  all  admitted  and 
made  use  of  the  following  law: 

The  quantity  of  heat  set  free  by  a  system  which  undergoes  a  trans- 
formation depends  solely  upon  the  initial  and  final  states  of  the  system 
and  not  at  all  upon  the  intermediate  states. 

This  law  may  also  be  stated  as  follows: 

When  a  system  passes  through  a  closed  cycle,  the  liberation  and 
absorption  of  heat  are  so  compensated  that  the  total  quantity  of  heat 
set  free  is  equal  to  0. 

A  reasoning  similar  to  that  used  in  Art.  21  would  prove  the 
equivalence  of  these  two  statements. 

It  is  easy  to  see  that  this  law  is  not  compatible  with  the  princi- 
ple of  the  equivalence  between  heat  and  work. 

Let  us  cause  a  system  to  pass  from  an  initial  state  0  to  a  final 
state  1  and  suppose  it  at  rest  in  both  of  these  states;  we  shall  then 
have  TF0=0,  1^=0,  and,  by  equation  (4)  of  the  preceding  chapter, 
the  quantity  of  heat  set  free  will  be 

(1)  Q=  U0-Ut+^. 

The  difference  (U^—U^  has  a  value  which  depends  exclusively 
upon  the  initial  and  final  states  of  the  system.  But  in  general 

W 
this  is  not  so  for  Wf,  nor  for  -^,  because  (Arts.  9  and  10)  the 

36 


CHEMICAL  CALORIMETRY.  37 

external  forces  which  act  on  the  system  do  not  in  general  admit 
a  potential.  We  must  then,  contrarily  to  the  preceding  law,  an- 
nounce the  following  proposition; 

The  quantity  of  heat  set  free  by  a  system  that  undergoes  a  trans- 
formation does  not  depend  solely  upon  the  initial  and  final  states 
but  also  upon  all  the  steps  of  the  transformation. 

32.  Example  from  the  study  of  perfect  gases.—  Let  us  con- 
sider immediately  an  example  of  this. 

M  grammes  of  a  perfect  gas  are  taken  at  the  temperature  t0, 
under  the  pressure  P;  they  occupy  a  volume  F0.  At  the  constant 
pressure  P  this  mass  of  gas  is  heated  to  the  temperature  t1}  higher 
than  t0;  it  then  occupies  a  volume  Vl  greater  than  F0;  at  the  same 
time  there  is  set  free  the  quantity  of  heat 

Q=-MC(tl-t0). 

This  gas  may  be  brought  from  the  same  initial  to  the  same 
final  state  by  another  process,  as  follows  : 

1°.  It  is  heated,  at  the  constant  volume  Y0,  from  the  tempera- 
ture t0  to  the  temperature  tv  during  which  operation  it  absorbs 
a  quantity  of  heat  Mcfa—  t0). 

2°.  The  reservoir  of  volume  V0,  containing  the  gas,  is  put  in 
communication  with  an  empty  reservoir  of  volume  (V1—  V0)  and 
the  temperature  is  allowed  to  return  to  ^;  according  to  Gay- 
Lussac's  experiments,  this  operation  occasions  neither  liberation 
nor  absorption  of  heat. 

The  second  transformation  therefore  produces  a  total  libera- 
tion of  heat, 


Although  these  two  transformations  bring  the  system  from 
the  same  initial  to  the  same  final  state,  they  do  not  give  the  same 
liberation  of  heat;  we  have,  in  fact, 

Q'-Q  =  M(C-c)(t1-t0), 

or,  according  to  Robert  Mayer's  relation   [equation  (14)  of   the 
preceding  chapter], 


38  THERMODYNAMICS  AND  CHEMISTRY. 

33.  Case  in  which  the  quantity  of  heat  set  free  by  a  system 
depends  solely  upon  the  initial  and  final  states.  —  The  law  stated 
at  the  beginning  of  Art.  31  is  then  in  general  false;  nevertheless 
there  are  particular  cases  in  which  it  is  true. 

Let  us  take  equation  (1),  which  gives  us  the  quantity  of  heat 
set  free  by  a  system  when  it  passes  from  a  state  0  in  which  its 
kinetic  energy  is  zero  to  a  state  1  in  which  its  kinetic  energy  is  also 
zero.  In  order  that  the  value  of  this  quantity  may  depend  solely 
upon  the  initial  and  final  states  and  in  no  wise  upon  the  inter- 
mediate states,  it  is  necessary  and  sufficient  that  the  same  is  true 
of  W  e]  in  other  terms  (Arts.  9  and  10),  in  order  that  the  quantity 
of  heat  set  free  by  a  system  which  undergoes  transformation  may 
depend  solely  upon  the  initial  and  final  states,  it  is  necessary  and 
sufficient  that  the  external  forces  which  act  upon  the  system  admit  a 
potential. 

If  Q  is  this  potential,  we  have  [chap.  I,  eq.  (12)] 


and  equation  (1)  becomes 

(2)  (Hff.  +  ^-^-J. 

The  systems  studied  by  the  chemist  may  be  regarded,  in  most 
cases,  as  undergoing  a  single  external  action,  that  of  a  normal, 
uniform  pressure;  this  pressure  (Art.  9)  does  not  in  general  admit 
a  potential;  nevertheless,  we  may  impose  upon  the  transforma- 
tions of  the  system  studied  restrictions  such  as  were  admissible 
in  Art.  12;  it  is  thus  in  the  two  following  particular  cases: 

1°.  The  external  pressure  remains  a  constant  value  P.  The 
potential  of  the  external  actions  is  then  [Chap.  I,  eq.  (14)]  Q  =  PV, 
and  equation  (2)  becomes 

PV  PV 

(3)  Q=i70  +  ^-C71-^i. 

2°.  The  volume  occupied  by  the  system  rests  constant.  The  ex- 
ternal pressure  then  admits  the  potential  (Art.  12)  £=0  and 
equation  (2)  becomes 


CHEMICAL  CALORIMETRY.  39 

The  chemist  has  then  the  right  to  use  the  law  that  the  early 
thermo-chemists  regarded  as  general,  when  they  make  use  of  one 
or  the  other  of  the  two  particular  cases  that  we  have  just  defined; 
they  are,  happily,  the  cases  which  are  the  most  common  in  investi- 
gations. 

1°.  Very  often  all  the  transformations  of  the  system  studied 
take  place  in  an  open  calorimeter,  that  is  to  say,  under  atmospheric 
pressure;  this  last  being  practically  constant,  the  first  of  the  above 
conditions  is  satisfied. 

2°.  Very  often,  also,  all  the  modifications  of  the  system  studied 
are  produced  inside  of  the  same  combustion-chamber  or  within 
the  same  calorimetric  bomb;  during  such  transformations  the 
volume  occupied  by  the  system  does  not  change,  so  that  the 
second  condition  is  fulfilled. 

34.  Utility,  in  chemical  calorimetry,  of  the  preceding  law.  — 
In  all  cases  in  which  the  law  stated  in  the  preceding  article  is  appli- 
cable it  renders  very  great  services  to  chemical  calorimetry. 

Suppose  that  a  system  sets  free  a  quantity  of  heat  q  during  a 
certain  modification  m  which  brings  it  from  the  state  0  to  the  state 
1,  a  quantity  of  heat  Q  during  a  modification  M  which  brings  it 
from  the  state  1  to  the  state  2,  finally  a  quantity  of  heat  Q'  during 
a  modification  M'  that  brings  it  from  the  state  0  to  the  state  2. 

Suppose,  besides,  that  the  modifications  m,  M,  M'  are  accom- 
plished in  such  conditions  that  the  external  forces  admit  a  poten- 
tial; for  example,  if  the  external  forces  reduce  to  a  normal,  uni- 
form pressure,  suppose  the  three  modifications  m,  M,  M'  accom- 
plished either  under  the  same  pressure  or  at  the  same  volume. 

The  series  of  modifications  m  and  M  on  the  one  hand,  the 
modification  M'  on  the  other  hand,  bring  the  system  from  the 
same  initial  state  0  to  the  same  final  state  2;  each  should  set  free 
the  same  quantity  of  heat 


(5) 

or 

(6)  q=Q'-Q, 

Now  it  may  happen  that  the  modification  M  does  not  readily 
admit  of  calorimetric  determinations,  while  the  modifications  M  and 


40  THERMODYNAMICS  AND  CHEMISTRY. 

M' ,  on  the  contrary,  are  easily  produced  within  a  calorimeter.  The 
measurement  of  the  two  quantities  of  heat  Q  and  Qf  joined  to 
equation  (6)  will  give  the  data  for  the  determination  of  the  heat  q. 
,  Similarly,  if  the  modifications  m,  M  are  adapted  to  calorimetric 
measurements  while  the  modification  M'  is  not  so  adapted,  equa- 
tion (5)  allows  the  determination  of  the  quantity  of  heat  Q'  from 
the  measurement  of  the  quantities  q  and  Q. 

This  remark  was  first  made  by  Berthollet,  who  gave  the  follow- 
ing application : 

To  determine  the  quantity  of  heat  (  — Q')  absorbed  when  a 
certain  quantity  of  salt  melts  a  certain  quantity  of  ice  at  atmos- 
pheric pressure.  This  modification  M'  causes  the  system  to  pass 
from  the  state  0,  in  which  the  ice  exists  separately,  at  0°,  to  the 
state  2,  composed  of  a  solution  brought  also  to  0°. 

Under  atmospheric  pressure  let  us  melt  the  mass  of  ice  con- 
sidered, the  salt  remaining  isolated;  this  modification  m  causes 
the  system  to  pass  from  the  state  0  to  the  state  1,  formed  of  liquid 
water  and  of  salt  separated  from  one  another  and  both  at  0°;  we 
can  measure  the  quantity  of  heat  that  the  ice  absorbs. 

Still  at  atmospheric  pressure,  dissolve  the  salt  in  water;  this 
modification  M  causes  the  system  to  pass  from  the  state  1  to  the 
state  2;  it  may  be  easily  produced  in  a  calorimeter,  so  that  the 
quantity  of  heat  absorbed  ( —  Q)  may  be  measured. 

Whence,  by  equation  (5),  the  quantity  of  heat  sought  (  —  Q') 
may  be  found;  the  slowness  of  the  modification  M'  would  not 
have  permitted  its  determination  directly. 

Let  us  take  an  application  of  equation  (6) ;  it  applies  to  cases 
in  which  all  the  modifications  of  the  system  take  place  in  a  reser- 
voir of  constant  volume. 

Suppose  that  12  grammes  of  diamond  (C)  at  0°  are  in  the 
presence  of  16  grammes  of  oxygen  (02);  this  is  the  state  0  of  the 
system. 

By  an  incomplete  combustion  (modification  m)  the  diamond 
combines  with  16  grammes  of  oxygen,  forming  the  mixture  CO  +  0, 
brought  to  0°,  which  is  the  state  1 ;  it  is  desired  to  know  the  quan- 
tity of  heat  q  set  free  by  this  reaction;  this  cannot  be  done  directly, 
because  it  is  impossible  to  regulate  the  combustion  so  that  the 
product  corresponds  exactly  to  the  preceding  formula. 


CHEMICAL  CALORIMETRY.  41 

But,  as  was  done  by  Berthelot  and  Matignon,  the  two  following 
reactions  may  be  realized  in  a  calorimetric  bomb : 

1°.  The  complete  combustion  of  the  oxide  of  carbon  (modifi- 
cation M)  which  causes  the  system  to  pass  from  the  state  1,  formed 
by  the  mixture  of  28  grammes  of  carbon  monoxide  and  16  grammes 
of  oxygen  (CO-f  O),  to  the  state  2,  formed  by  44  grammes  of  carbon 
dioxide  (CO2)  brought  to  0°.  This  modification  sets  free  a  quantity 
of  heat 

0=68200  calories. 

2°.  The  complete  combustion  of  the  diamond  (modification 
M ')  which  causes  the  system  to  pass  from  the  state  0  to  the  state 
2.  This  modification  sets  free  a  quantity  of  heat 

Q'  =  94300  calories. 

Equation  (6)  is  here  applicable  and  gives 
q =26100  calories. 

35.  Exothermic  and  endothermic  reactions. — Here  is  an- 
other application  of  equation  (6),  made  in  1852  by  Favre  and 
Silbermann. 

The  state  0  of  the  system  is  formed  by  12  grammes  of  carbon 
(C)  and  88  grammes  of  gaseous  nitrous  oxide  (2N2O),  at  the  tem- 
perature 0°.  Without  change  of  volume,  pass  to  state  1  (modifi- 
cation ra)  formed  by  12  grammes  of  carbon  in  the  presence  of  a 
mixture  of  56  grammes  of  nitrogen  and  32  grammes  of  oxygen 
(4N+2O),  the  whole  brought  to  0°.  The  modification  m  sets  free 
a  quantity  of  heat  q  which  it  is  proposed  to  find. 

For  this  we  determine: 

1°.  The  quantity  of  heat  Q  set  free  by  the  combustion  (modi- 
fication M)  of  12  grammes  of  carbon  in  the  mixture  4N+20; 
the  state  2  of  the  system  is  formed  by  56  grammes  of  nitrogen  .and 
44  grammes  of  carbonic  acid  gas  (4N+C02),  brought  to  0°. 

2°.  The  quantity  of  heat  Q'  set  free  by  the  combustion  (modi- 
fication M')  of  12  grammes  of  carbon  in  88  grammes  of  nitrous 
oxide;  the  system  passes  from  the  state  0  to  the  state  2. 

All  the  experiments  of  Favre  and  Silbermann  were  performed 
at  atmospheric  pressure. 

q  is  then  given  by  equation  (6). 


42  THERMODYNAMICS  AND  CHEMISTRY. 

Now,  the  measurements  of  Favre  and  Silbermann  have  shown 
that  the  quantity  Q'  was  greater  than  the  quantity  of  heat  Q; 
the  quantity  of  heat  q  is  therefore  positive;  thus,  in  the  presence 
of  a  mass  of  carbon  which  takes  no  part  in  the  modification  m  and 
of  which,  consequently,  it  is  not  necessary  to  take  account,  the 
decomposition  of  nitrous  oxide  into  nitrogen  anh  oxygen  liberates 


This  result  caused  surprise  at  the  epoch  when  Favre  and  Silber- 
mann obtained  it;  up  to  that  time,  in  fact,  it  was  believed  that 
every  chemical  combination  liberated  heat,  and  that  every  chemi- 
cal decomposition  absorbed  heat. 

Since  this  period,  Favre  and  Silbermann  's  observation  has  been 
accurately  repeated;  by  using  the  calorimetric  bomb  and  operating, 
consequently,  at  constant  volume,  Berthelot  has  made  the  follow- 
ing measurements: 

1°.  Modification  M': 


=  C02+2N, 
Q'  =  88800  calories. 


2°.  Modification  M: 


=  C02+2N, 
Q  =  68200  calories. 

It  is  concluded  that  the  modification  m, 

CO  +  N2O  =  CO+2N+0, 
sets  free  a  quantity  of  heat 

q  =20600  calories. 

Also,  for  reasons  that  would  be  too  long  to  discuss  here,  it  is 
admitted'  that,  in  this  last  reaction,  the  presence  of  the  carbonous 
oxide  may  be  neglected,  as  it  takes  no  part  in  the  reactions,  so  that 
the  decomposition,  at  constant  volume,  of  44  grammes  of  nitrous 
oxide  liberates  20600  calories. 

There  are  many  examples  of  the  liberation  of  heat  by  chemical 
decomposition.  Let  us  cite  two  taken  from  among  the  determina- 
tions made  by  Berthelot  by  means  of  the  calorimetric  bomb. 


CHEMICAL  CALORIMETRY.  43 

First  example:   Decomposition  of  nitric  oxide. 
Operation  M: 

2CN+40+4N  =  2C02+6N, 
Q= 261800  calories. 

Operation  M': 

2CN+4NO  =  2CO2+6N, 
Q'  =  349200  calories. 

Operation  m: 

2CN+4NO  =  2CN+4N+4O, 

q =87400  calories. 

Thus  30  grammes  of  nitric  oxide,  in  decomposing  at  constant 
volume,  set  free  ^4^  =  21850  calories. 

Second  example:   Decomposition  of  acetylene. 
Operation  Mr: 

C2H4  +  5O  =  2CO2  +  H2O  liquid, 
Q'  =  314900  calories. 

Operation  M : 

2C + 2H  +  5O  =  2CO2  +  H2O  liquid, 
Q = 94300  X  2  +  69000  =  257600  calories. 

Operation  m: 

C2H2  +  5O  =  2C + 2H  + 5O, 
q  =  57300  calories. 

Thus  26  grammes  of  acetylene,  decomposing  at  constant  vol- 
ume, liberate  57300  calories. 

When,  in  certain  circumstances,  the  formation  of  a  compound 
from  its  elements  liberates  heat,  or  when  its  decomposition  absorbs 
heat,  the  compound  is  called  an  exothermic  compound  in  the  cir- 
cumstances considered. 

Water,  hydrochloric  acid,  carbonous  oxide,  carbonic  anhydride, 
are,  in  all  circumstances,  exothermic  compounds. 


44  THERMODYNAMICS  AND  CHEMISTRY. 

When,  in  certain  circumstances,  a  compound  is  formed  with 
absorption  of  heat  or  is  decomposed  with  liberation  of  heat,  it  is 
with  an  endothermic  compound  that  one  has  to  deal. 

Nitrous  oxide,  nitric  oxide,  acetylene  are  endothermic  in  all 
circumstances. 

36.  Heats  of  formation  at  constant  pressure  and  at  constant 
volume. — Imagine  that  a  mixture  of  two  bodies  A  and  B  com- 
bine and  furnish  1  gramme  of  the  compound  C.  The  quantity 
of  heat  set  free  by  this  combination  depends  upon  the  conditions 
in  which  it  is  produced. 

Suppose,  in  the  first  place,  that  during  the  combination  the 
temperature  t  remains  constant  and  that  the  pressure  also  has 
a  constant  value  P;  let  L  be  the  heat  set  free  during  this  combina- 
tion; L  is,  at  the  temperature  t  and  constant  pressure  P,  the  heat 
of  formation  of  the  compound  C. 

Suppose,  in  the  second  place,  that  during  the  combination  the 
temperature  t  and  the  volume  V  occupied  by  the  system  are  kept 
constant;  let  A  be  the  heat  set  free  during  the  combination;  ^  is 
called  the  heat  of  formation  of  the  compound  C,  at  the  temperature  t 
and  at  the  constant  volume  V. 

It  may  happen  that,  in  the  conditions  considered,  the  elements 
A  and  B  cannot  enter  into  combination,  but  on  the  contrary  the 
compound  C  decomposes  into  its  elements. 

If,  at  the  constant  temperature  t  and  at  the  constant  pressure 
P,  1  gramme  of  the  body  C  is  decomposed  into  its  elements  A  and 
B,  and  if  this  reaction  absorbs  a  quantity  of  heat  L  or  liberates  a 
quantity  —  L  of  heat,  L  is  still  called  the  heat  of  formation  of  the 
compound  C,  under  the  constant  pressure  P,  at  the  temperature 
t]  a  similar  remark  may  be  made  for  the  heat  of  formation  at  con- 
stant volume. 

From  these  definitions,  if  a  compound  C  is  exothermic  in  the 
given  conditions,  its  heat  of  formation  in  these  conditions  is  positive; 
it  is  negative  if  the  compound  C  is  endothermic. 

Often,  instead  of  considering,  in  the  preceding  definitions,  1 
gramme  of  compound  C,  w  grammes  are  considered,  w  being  the 
molecular  weight  of  the  compound  C.  The  quantities  L  and  X  are 
then  replaced  by  other  quantities  L'  and  I,  which  are  equal  to  wL 
and  wX  respectively.  These  quantities  L'  and  I  are,  at  the  tern- 


CHEMICAL  CALORIMETRY.  45 

perature  t,  the  molecular  heats  of  formation  of  the  compound  C, 
the  one  at  constant  pressure  P,  the  other  at  constant  volume  V. 

It  is,  in  general,  the  values  of  L'  and  I  which  are  found  in  ther- 
mochemical  tables. 

37.  Case  in  which  the  two  heats  of  formation  are  equal  to 
each  other. — If  the  combination,  taking  place  at  constant  tem- 
perature and  under  constant  pressure,  does  not  cause  the  volume 
of  the  system  to  change,  case  in  which  the  combination  takes  place 
without  contraction  or  expansion,  it  follows  from  the  definition  that 
the  two  heats  of  formation  L  and  X  are  equal  to  each  other. 

Thus,  at  a  given  temperature,  the  heat  of  formation  of  hydro- 
chloric acid  gas  under  constant  pressure  and  the  heat  of  formation 
of  this  gas  under  constant  volume  have  an  identical  value. 

38.  General  relation  between  the  two  heats  of  formation. — 
In  general  the  above  is  not  true  if  the  combination,  taking  place 
at  constant  temperature  and  under  constant  pressure,  causes  the 
volume  of  the  system  to  vary. 

Suppose  that  1  gramme  of  the  mixture  A+J5,  brought  to  the 
temperature  t  and  supporting  the  pressure  P,  occupies  the  volume 
F0;  that,  in  the  same  conditions,  1  gramme  of  the  compound  C 
occupies  the  volume  V^;  let,  in  these  conditions,  UQ  be  the  internal 
energy  that  1  gramme  of  the  mixture  A  +B  possesses,  Vl  the  in- 
ternal energy  of  1  gramme  of  the  compound  C. 

Equation  (3)  gives 

(7)  L=U0-Ul  +  ^(V0-Vl). 

Also,  denote  by  uv  the  internal  energy  of  1  gramme  of  the  com- 
pound C,  at  the  temperature  and  the  volume  F0.  Equation  (4) 
gives 

(8)  X^U.-u,. 
Therefore  we  have 

(9)  L-^=ul-Ul  +  ^(Va-V1). 

Such  is,  in  general,  the  expression  for  the  difference  which 
exists  between  the  heat  of  formation  at  constant  pressure  and 
the  heat  of  formation  at  constant  volume  of  the  same  compound, 


46  THERMODYNAMICS  AND  CHEMISTRY. 

at  the  same  temperature.  This  difference  depends  upon  the 
diminution  which  the  internal  energy  of  1  gramme,  undergoes 
when,  without  varying  the  temperature  t,  the  volume  changes 
from  VQ  to  Fj. 

39.  Case  in  which  the   compound  is  a  perfect  gas. — This 
variation  of  energy  is  in  general  not  known;   but  there  is  a  par- 
ticular case  in  which  we  know  how  to  evaluate  this  variation;   it 
is  the  case  in  which  the  compound  C  is  a  gas  near  enough  to  the 
perfect  state  that  we  may  apply  the  laws  which  characterize  this 
state;    in  this  case,  the  two  internal  energies  u1}   Ul  refer  to  the 
same  mass  of  gas,  taken  at  the  same  temperature  t,  consequently 
(Art.  25)  they  are  equal  to  each  other  and  equation  (9)  reduces  to 

(10)  L-J-JC7.-7J. 

When  the  compound  is  a  perfect  gas,  the  excess  of  the  heat  of  for- 
mation at  constant  pressure  over  the  heat  of  formation  at  constant 
volume  is  equivalent  to  the  external  work  done  by  the  formation  of  a 
gramme  of  the  compound  under  the  constant  pressure  considered. 

This  proposition  assumes  nothing  as  to  the  nature  of  the  com- 
ponents, which  may  be  solid,  liquid,  or  gas,  the  latter  being  or  not 
near  the  ideal  state. 

40.  The  distinction  between  the  two  heats  of  formation  has 
small  importance  in  practice. — Let  us  apply  the  preceding  for- 
mulas to  the  calculation  of  the  difference  between  the  two  heats 
of  formation  of  water  vapor,  both  referred  to  0°;  the  pressure  P 
is  taken  equal  to  atmospheric  pressure. 

Let  us  take  for  unit  of  force  the  gramme-weight,  for  unit  of 
length  the  centimetre,  for  unit  of  heat  the  small  calorie;  we  shall 
then  have 

P=  1033.3, 

#=42500. 

The  volume  of  1  gramme  of  water  vapor,  in  the  normal  condi- 
tions of  temperature  and  pressure,  reckoned  in  cubic  centimetres, 
has  the  value 

F.         -1- 

1    0.622X0.001293' 


CHEMICAL  CALORIMETRY.  47 

Again,  water  vapor  is  formed  with  a  contraction  equal  to  J,  so 
that  we  have 


3' 
or 


We  then  find 

L—  ^=15.1  calories. 

If  we  observe  that  ^  is  equal  to  about  3220,  we  see  that  the 
difference  between  these  two  heats  of  formation  of  water  vapor 
is  negligible  compared  with  each  of  these  two  heats  of  formation. 

It  is  thus  for  the  greater  number  of  cases.  The  distinction 
between  the  heat  of  formation  at  constant  pressure  L  and  the  heat 
of  formation  at  constant  volume  X,  essential  from  a  theoretical 
point  of  view,  has,  in  general,  a  minimum  of  practical  importance. 

41.  Influence  of  temperature  on  the  heats  of  formation.  — 
Of  greater  practical  importance  are  the  following  remarks: 

In  defining  the  heats  of  formation  of  a  given  compound  at 
constant  pressure  and  at  constant  volume,  we  have  stated  the 
temperature  at  which  the  reaction  takes  place.  This  indication 
is  essential,  for  the  two  quantities  L  and  X  vary,  in  general,  with 
the  temperature,  and  we  shall  now  discuss  the  laws  of  these  varia- 
tions. 

Let  us  consider,  for  example,  the  heat  of  formation  at  constant 
pressure. 

Take  1  gramme  of  the  mixture  A  +  B  under  the  constant  pres- 
sure P  at  the  temperature  t  and,  without  varying  either  pressure 
or  temperature,  cause  the  compound  C  to  be  formed  ;  the  system 
liberates  a  quantity  of  heat  I/;  next  bring  the  compound  C  from 
the  temperature  t  to  the  temperature  t'\  it  absorbs  a  quantity  of 
heat  C(t'  —  t\  C  being  the  -mean  specific  heat  of  the  compound  C, 
between  the  temperatures  t  and  t',  under  the  constant  pressure  P. 
The  total  quantity  of  heat  liberated  in  the  transformation  con- 
sidered has  the  value 

L-C(t'-t). 


48  THERMODYNAMICS  AND  CHEMISTRY. 

Now  take  1  gramme  of  the  mixture  A +B,  under  the  constant 
pressure  P;  without  letting  it  undergo  any  combination,  bring  it 
from  the  temperature  t  to  the  temperature  t' ;  it  absorbs  a  quantity 
F(t'  —  t)  of  heat,  F  being  the  specific  heat  of  the  mixture  at  the 
constant  pressure  P;  then  at  the  temperature  t'  and  the  pressure 
P,  cause  the  mixture  to  combine;  it  liberates  a  quantity  of  heat 
//.  The  total  quantity  of  heat  liberated  by  this  second  trans- 
formation is 

L'-r(t'-t). 

The  two  transformations  cause  the  system  to  pass  from  the 
same  initial  to  the  same  final  states;  they  take  place  at  the  same 
pressure  P;  they  therefore  liberate  the  same  quantities  of  heat 
and  we  have 

(11)  L'-L=(r-C)(t'-t). 

A  similar  line  of  reasoning  may  be  applied  to  the  heat  of  forma- 
tion at  constant  volume;  if  A  and  X'  are  the  values  of  this  heat  at 
the  temperatures  t  and  t',  if  7-  and  c  -are  the  mean  specific  heats 
between  the  temperatures  t  and  t',  at  constant  volume,  of  the  mix- 
ture A+B  and  of  the  compound  C,  we  have 

(12)  x-x=(r-c}(t>-t). 

42.  Heat  of  formation  referred  to  a  temperature  at  which 
the  reaction  considered  is  impossible. — It  often  happens  that  in 
books  on  thermochemistry  the  heat  of  formation  of  water  at  0° 
is  spoken  of,  while  at  0°  oxygen  and  hydrogen  will  not  combine 
and  water  is  undecomposable ;  the  definitions  of  the  quantities  L 
and  A  are  therefore  illusory  at  this  temperature  and  the  words 
used  seem  void  of  sense. 

We  may  interpret  them  in  the  following  way: 

Suppose  one  of  the  two  reactions  possible  at  the  temperature 
t;  at  this  temperature  the  two  quantities  L  and  A  have  the  ex- 
perimental meaning  that  we  have  given  them. 

If,  at  the  temperature  t',  the  two  reactions  are  impossible,  we 
shall  regard  the  two  heats  L'  and  X'  at  this  temperature  as  purely 
algebraical  quantities  defined  by  equations  (11)  and  (12). 


CHEMICAL  CALORIMETRY  49 

43.  Importance  of  the  variations  that  changes  of  tempera- 
ture cause  in  the  heats  of  formation. — It  follows  from  the  formulae 
(11)  and  (12)  that  changes  of  temperature  may  cause  very  consider- 
able variations  in  the  heats  of  formation  of   a  compound.     Ac- 
cording to  Berthelot  the  following  variations  exist  in  the  heat  of 
formation  of  water  vapor  under  atmospheric  pressure: 

at  + 15°  C.,       L=3228  calories 
2000°  2811 

4000°  2001       " 

We  see  that  changes  of  temperature,  provided  they  are  suffi- 
ciently extended,  may  cause  the  heat  of  formation  of  a  body  to 
vary  by  an  amount  comparable  with  the  value  of  this  heat. 

We  shall  even  meet  cases  in  which  a  body,  formed  with  ab- 
sorption of  heat  at  one  temperature,  is  formed  with  liberation  of 
heat  at  a  higher  temperature;  the  variations  of  temperature  in 
such  cases  change  the  sign  of  the  heat  of  formation  of  the  com- 
pound. 

44.  Case  of  perfect  gases  which  combine  without  conden- 
sation.    Delaroche  and  Be"rard's  law.     The  heats  of  formation 
are  independent  of  the  temperature.— The  general  formulae  (11) 
and  (12)  take  on  a  form  easier  to  apply  in  the  particular  cases  in 
which  the  two  mixed  bodies  A  and  B  are  perfect  gases.     In  this 
case  the  specific  heat  at  constant  pressure  of  the  mixture  is  ob- 
tained by  applying  the  classic  rule  of  mixtures  to  the  specific  heats 
at  constant  pressure  of  the  two  mixed  gases;   an  analogous  rule 
applies,  also,  to  the  specific  heats  at  constant  volume  of  the  mix- 
ture and  of  the  mixed  gases. 

Let  us  suppose  that  a  molecule  of  the  compound  C  is  formed  by 
the  union  of  na  molecules  of  the  body  A  and  nb  molecules  of  the 
body  B;  let  wa  and  w^  be  the  molecular  weights  of  these  two  bodies 
A  and  B,  and  w  the  molecular  weight  of  the  compound  C.  We 
have 

w=nawa+nbwb. 

In  order  to  form  1  gramme  of  the  compound  C,  it  would  be 
necessary  to  take  -^— -  grammes  of  the  body  A  and  — — -  grammes 


50  THERMODYNAMICS  AND  CHEMISTRY. 

of  the  body  B.  If  we  denote  by  Ca,  ca  the  two  specific  heats  of 
the  gas  A,  by  Cb,  cb  the  two  specific  heats  of  the  gas  B,  the  rule 
that  we  have  just  recalled  gives 


_nawa       nbwb 

*f  —  -  Ca  "I  ---  Cb. 

w  w 

Equations  (11)  and  (12)  may  be  written 

(13)  w(L'-L)  =  (nawaCa+nbwbCb-wC)(t'-t)', 

(14)  w(  Xf  —  X)  =  (nawaca  +  nbwbcb  —  we)  (tf  —  t)  . 

It  is  evident  that  these  equations  allow  of  determining  very 
simply  the  variation  of  the  heat  of  formation  with  the  tempera- 
ture, either  at  constant  pressure  or  at  constant  volume,  of  a  com- 
pound and  of  its  gaseous  components.  Here  is  a  remarkable 
example  : 

Suppose  that  the  body  C  is  a  sensibly  perfect  gas  formed  by  the 
union,  in  equal  volumes  and  without  condensation,  of  two  simplef 
diatomic,  sensibly  perfect  gases,  A  and  B. 

For  this,  hydrochloric  acid  gives  us  a  fair  example. 

In  this  case  a  molecule  of  the  compound  includes  a  half-mole- 
cule of  each  of  the  component  gases;  na,  nb  are  both  equal  to  £, 
and  equations  (13)  and  (14)  may  be  written 


(15) 
(16) 


Again,  a  very  old  law,  discovered  by  Delaroche  and  Be"rard, 
and  since  verified  by  many  observers,  shows  that  for  all  the  simple 
diatomic  gases,  near  to  the  ideal  state  and  for  all  gaseous  compounds, 
formed  without  condensation  and  near  to  the  ideal  state,  the  product 


CHEMICAL  CALORIMETRY. 


51 


of  the  molecular  weight  by  the  specific  heat  at  constant  pressure  has  a 
single  value: 

(17)  Wa,Ca  =  WbCb  —  W<7. 

Here  are  some  examples  of  the  truth  of  this  law,  taken  from 
Regnault's  observations: 


1°  Simple  diatomic  gases. 

Value  of  wXC 

Oxygen     .              ... 

3  4800 

Nitrogen  

3  4112 

Hydrogen.  

3  4128 

2°  Gaseous  compounds  formed  without 
condensation. 

Value  of  wXC 

Nitric  oxide  

3.4800 

Carbon  monoxide 

3  4128 

Hydrochloric  acid  

3.3744 

If  we  denote  by  oa,  Ob,  &  the  volumes  occupied,  in  the  normal 
conditions  of  temperature  and  pressure,  by  one  gramme  of  each 
of  the  gases  A,  B,  C,  respectively,  Robert  Mayer's  relation  [Chap. 
II,  eq.  (14)]  gives 


•p 


W(C  —  C)  =  ^,W<7. 
til 

Again,  from  Avogadro's  Law,  we  have 


so  that  we  may  write 

wa(Ca-ca)=wb(Cb-cb)=w(C-c). 
From  equation  (17)  it  follows  that 
(18)  waca  =  wycb  =  we. 


52  THERMODYNAMICS  AND  CHEMISTRY. 

The  law  of  Delaroche  and  Berard  applies  also  to  the  sp.cific  heats 
at  constant  volume. 

By  means  of  equations  (17)  and  (18),  equations  (15)  and  (16) 
become 

j 

(19)  L-L'=0; 

(20)  A-A'=0. 

When  a  perfect  gas  is  formed  by  the  union  in  equal  volumes  and 
without  condensation  of  two  simple,  diatomic  gases,  the  heat  of  forma- 
tion at  constant  pressure  and  the  heat  of  formation  at  constant  volume 
(equal  to  each  other)  are  independent  of  the  temperature. 


CHAPTER  IV. 

CHEMICAL    EQUILIBRIUM   AND    THE   REVERSIBLE   TRANS- 
FORMATION. 

45.  Idea  of  chemical  equilibrium.    It  differs  from  the  idea 
of  mechanical  equilibrium. — In  mechanics  a  system  of  bodies  is 
said  to  be  in  equilibrium  when  each  of  these  bodies  and  each  of 
the  parts  that  compose  it  keep  an  invariable  form  and  position 
in  space. 

In  chemistry  a  system  is  said  to  be  in  chemical  equilibrium 
when  no  chemical  reaction  is  taking  place  in  it. 

Chemical  equilibrium  is  not  a  special  case  of  mechanical  equi- 
librium. It  is  possible  to  observe  a  chemical  reaction  in  a  system 
notwithstanding  that  each  of  the  parts  of  the  system  keeps  an 
invariable  form  and  position.  Such  a  system  is  therefore  in  me- 
chanical equilibrium,  but  not  in  chemical  equilibrium. 

Let  us  take,  for  example,  a  homogeneous  mixture  of  hydrogen 
and  chlorine  and  let  diffused  light  act  upon  it  in  a  hermetically 
closed  reservoir;  this  mixture  remains  perfectly  at  rest;  each  part, 
however  small,  as  far  as  can  be  distinguished  keeps  an  invariable 
position  and  form;  yet  the  system  is  the  seat  of  a  chemical  reac- 
tion; the  hydrogen  and  chlorine  combine  to  form  hydrochloric 
acid  gas. 

46.  Chemical  equilibrium  may  be  the  common  limit  of  two 
oppositely  directed  reactions.     Phenomena  of  saponification. — 
In  the  example  that  we  have  just  cited,  the  reaction  does  not  cease 
until  one  of  the  gases,  chlorine  or  hydrogen,  which  takes  part  in 
the  reaction  has  entirely  disappeared.     The    reaction    does  not 
stop  until  the  instant  that  it  would  be  absurd  to  suppose  that  it 
persisted;  such  a  reaction  is  called  an  unlimited  reaction. 

53 


54  THERMODYNAMICS  AND  CHEMISTRY. 

It  is  not  so  with  all  reactions  in  chemistry.  For  example, 
Berthelot  and  Pean  de  Saint-Giles  *  mixed  together  masses  of 
water  and  benzoic  ether  proportional  to  ;the  molecular  weights  of 
these  two  substances;  they  heated  them  at  200°  in  a  sealed 
tube;  saponification  was  produced;  that  is  to  say,  there  was  forma- 
tion of  benzoic  acid  and  alcohol  according  to  the  equation 

C6H5CO2C2H5  +  H2O  =  C6H5C02H  +  C2H5OH. 

benzoic  ether  +  water  =  benzoic  acid  +  alcohol 

If  the  reaction  were  unlimited,  it  would  not  stop  so  long  as 
the  mixture  enclosed  a  quantity,  however  small,  of  benzoic  ether 
and  water. 

This  is  not  what  is  observed. 

At  the  end  of  24  hours'  heating,  the  mixture  still  encloses  a 
certain  mass  of  non-saponified  ether.  This  mass  is  a  considerable 
fraction  of  the  quantity  of  ether  introduced  originally  into  the 
sealed  tube,  the  fraction  being  0.664.  The  time  of  heating  may 
be  indefinitely  prolonged  without  producing  the  slightest  change 
in  the  composition  of  the  system;  the  saponification  has  there- 
fore ceased,  and  it  has  stopped  when  its  continuation  would  be 
nowise  in  contradiction  to  the  chemical  formulae,  and  while  there 
still  exists  in  the  system  bodies  susceptible  of  taking  part. 

This  fact  is  expressed  by  saying  that  the  saponification  of 
benzoic  ether  at  200°  is  a  limited  reaction. 

Now  let  us  mix,  as  did  Berthelot  and  Pean  de  Saint-Giles, 
masses  of  benzoic  acid  and  alcohol,  proportional  to  the  molecular 
weights  of  these  two  bodies;  heat  them  at  200°  in  a  sealed  tube; 
we  shall  observe  the  inverse  to  saponification,  a  formation  of 
benzoic  ether  and  water  represented  by  the  equation 


C.H5OH  =  C6H5C02C2H5  +  H20. 

id  +  alcohol  =  benzoic  ether  +  water 


benzoic  acid  + 

This  reaction,  the  inverse  of  the  preceding,  is,  like  it,  a 
limited  reaction;  it  stops  before  the  benzoic  acid  and  alcohol 
have  been  transformed  entirely  into  benzoic  ether  and  water;  no 
matter  how  long  the  experiment  is  prolonged,  the  mass  of  ben- 
zoic ether  obtained  remains  equal  to  a  fraction  of  the  mass  that 

1  BERTHELOT  and  PEAN  DE  SAINT-GILES,  Annales  de  chimie  et  de  Physique, 
v.  45,  p.  485,  1862;  v.  46,  p.  5,  1862;  v.  48,  p.  225,  1863. 


CHEMICAL  EQ&J&B&I&ti&r  55 


would  be  obtained  if  the  reaction  were  unlimited;  this  fraction  is 
0.664. 

Let  us  compare  these  two  experiments. 

In  both,  the  starting-point  was  two  mixtures  whose  elemen- 
tary composition  is  the  same;  we  may  regard  these  two  mixtures 
as  the  two  extreme  states  of  the  same  system;  one,  the  mixture 
of  ether  and  water,  represents  the  extreme  state  of  etherification ; 
the  other,  the  mixture  of  alcohol  and  acid,  represents  the  state 
of  extreme  saponification. 

Starting  from  these  two  different  states,  there  is  produced  at 
200°  two  reactions  the  inverse  of  each  other;  in  the  midst  of  the 
completely  etherified  system  a  saponification  is  produced;  in  the 
midst  of  the  completely  saponified  system  an  etherification  is 
produced.  Each  of  these  two  reactions  is  limited.  Each  of 
them  ceases  when-  the  mixture  attains  a  certain  composition  in- 
termediate between  total  etherification  and  total  saponification. 
This  composition  for  which  chemical  equilibrium  is  established 
is  the  same  in  the  two  cases.  It  is  obtained  when  the  mass  of 
ether  existing  in  the  system  is  a  fraction  of  the  mass  of  possible 
ether,  equal  to  0.664. 

Thus,  at  200°,  chemical  equilibrium  is  established  in  the  sys- 
tem considered  when  the  mass  of  ether  that  it  encloses  is  the  frac- 
tion 0.664  of  the  possible  mass  of  ether.  This  state  of  chemical 
equilibrium  is  the  common  limit  of  two  inverse  reactions,  etherifica- 
tion and  saponification. 

47.  Reciprocal  action  of  two  soluble  salts  within  a  solu- 
tion.— The  phenomena  of  etherification,  studied  by  Berthelot 
and  Pean  de  Saint-Giles,  are  not  the  only  ones  in  which  may  be 
observed  a  state  of  chemical  equilibrium,  the  common  limit  of  two 
reactions  which  are  the  inverse  of  each  other.  Berthollet  had 
predicted  that  such  a  state  of  equilibrium  should  be  produced  in 
a  solution  in  which  two  soluble  salts  may,  by  double  decomposi- 
tion, produce  two  other  soluble  salts.  Malaguti  *  has  verified  the 
truth  of  Berthollet's  prediction  in  the  following  way : 

In  a  known  mass  of  water,  dissolve  a  molecule  of  strontium 
acetate  and  two  molecules  of  potassium  nitrate;  there  will  be 


1  MALAGUTI,  Annales  de  Chimie  et  de  Physique,  3me  s£rie,  v.  37,  1853. 


56  THERMODYNAMICS  AND  CHEMISTRY. 

formed,  in  the  solution,  potassium  acetate  according  to  the  equa- 
tion 

Sr(CH3C03)2 + 2KNO3  =  2KC2H3C02  +  Sr(NO3)2. 

To  determine  the  composition  of  the  mixture  at  a  given  in- 
stant, it  is  sufficient  to  treat  it  with  an  excess  of  alcohol  mixed 
with  ether;  this  alcohol  and  ether  dissolve  the  acetates,  but  not 
the  nitrates. 

If  this  analysis  is  made  after  a  long  wait  at  ordinary  tempera- 
ture, it  is  seen  that  the  double  decomposition  ceases  before  it  is 
completed;  the  condition  of  equilibrium  which  limits  this  double 
decomposition  corresponds  nearly  to  the  following  composition 
of  the  mixture: 

f  molecule  of  potassium  acetate, 
•§-  molecule  of  potassium  nitrate, 
J  molecule  of  strontium  nitrate, 
f  molecule  of  strontium  acetate. 

Suppose  now  that  there  is  dissolved  in  the  same  quantity  of 
water  one  molecule  of  strontium  nitrate  and  two  molecules  of 
potassium  acetate;  by  a  reaction  the  inverse  of  the  preceding, 
there  is  formed  strontium  acetate  and  potassium  acetate,  accord- 
to  the  equation 

Sr(NO3)2  +  2KC2H3C02 = Sr(CHoCO2)2  +  2KNO3. 

At  ordinary  temperatures  this  reaction  is  limited  and  leads 
to  the  same  state  of  equilibrium  as  the  preceding  reaction;  this 
state  of  equilibrium  is  therefore  again  the  common  limit  of  two 
reactions  that  are  the  inverse  of  each  other. 

48.  Many  chemical  systems  seem  incapable  of  possessing  a 
state  of  equilibrium  which  is  the  common  limit  of  two  recipro- 
cally inverse  reactions. — In  a  great  number  of  chemical  systems, 
states  of  equilibrium  similar  to  those  we  have  just  studied  are 
met;  each  of  these  states  of  equilibrium  is  the  common  limit  of 
two  reactions  the  inverse  of  each  other. 

But  a  not  less  number  of  chemical  systems  show  themselves, 
at  first  sight,  incapable  of  possessing  such  states  of  equilibrium. 
Let  us  take,  for  instance,  a  system  formed  of  oxygen  and  hydrogen; 


CHEMICAL  EQUILIBRIUM.  57 

if  we  are  content  to  observe  superficially  the  properties  of  this  sys- 
tem, we  shall  be  led  to  give  the  following  description,  for  a  long 
time  considered  the  true  one: 

At  low  temperatures  oxygen  and  hydrogen  do  not  combine; 
water  is  not  decomposed.  At  high  temperatures  oxygen  and 
hydrogen  combine;  this  combination  is  not  limited,  but  total; 
water  is  not  decomposable. 

49.  Grove's  experiment.     Water  is  decomposable  by  heat. — 
Notwithstanding,  an  old  experiment  contradicts  this  description 
of  the  properties  of  water.    By  letting  fall  into  water  a  sphere 
of  platinum  brought  to  white  heat,  an  explosion  is  produced; 
the  water  is  decomposed  by  the  contact  of  the  platinum  sphere; 
afterwards  the  oxygen  and  hydrogen  recombine. 

This  experiment,  due  to  Grove,  was  known  long  ago,  but  chem- 
ists were  content,  with  Berzelius,  to  attribute  it  to  the  catalytic 
farce  of  the  platinum.  This  experiment  was  repeated  on  a  larger 
scale  by  H.  Sainte-Claire  Deville  and  H.  Debray;  rejecting  the 
explanation  by  catalytic  force  and  accepting  purely  and  simply 
the  teaching  of  the  experiment,  they  admitted  the  following  propo- 
sition: At  a  temperature  inferior  to  the  fusing-point  of  platinum, 
water  vapor  is  decomposed  into  its  elements,  oxygen  and  hydrogen. 

Further,  water  is  even  decomposable  at  the  temperature  of 
the  fusing-point  of  silver,  that  is,  at  a  temperature  less  than  1000° 
C.  When  silver  is  melted  in  the  presence  of  water  vapor  it  ab- 
sorbs oxygen  and  gives  it  out  again  only  at  the  moment  of  solidi- 
fication, this  constituting  the  phenomenon  of  rochage;  the  rochage 
proves  therefore  that  the  elements  of  water  are  at  liberty  at  the 
temperature  of  1000°,  unless  one  would  attribute  to  melted  silver 
a  chemical  action  on  the  oxygen;  this  objection,  however,  may 
be  obviated;  it  is  sufficient  to  repkce  the  silver  by  litharge,  a 
substance  chemically  saturated  with  oxygen,  incapable  of  further 
oxidation;  the  phenomenon  of  rochage,  certain  proof  of  the  de- 
composition of  water,  is  produced  just  as  sharply. 

50.  Direct   demonstration   of    the   dissociation   of  water. — 
Other  experiments,  still  more  direct  and  more  conclusive,1  place 

1  H.  SAINTE-CLAIRE  DEVILLE,  Comptes  Rendus,  v.  56,  pp.  195  and  322, 
1863;  Lessons  on  Dissociation,  given  before  the  Chemical  Society,  Mar.  18 
and  April  1,  1864;  H.  DEBRAY,  Wurtfs  Dictionary,  art.  Dissociation. 


58  THERMODYNAMICS  AND  CHEMISTRY. 

beyond  doubt  the  dissociation  of  water  at  temperatures  easily 
attained  in  laboratories. 

Use  is  made  of  an  apparatus  composed  of  a  glazed  porcelain 
tube  VV  (Fig.  18),  in  the  interior  of  which  is  another  tube,  PP', 


FIG.  18. 

of  porous  substance;  the  two  tubes  being  strongly  heated  in  a 
furnace  fed  with  coke  or  hard  coal,  water  vapor  is  let  into  the 
interior  tube  of  porous  earthenware  at  t,  and  a  current  of  car- 
bonic acid  is  introduced  by  the  tube  6  into  the  annular  space 
included  between  the  porous  and  the  porcelain  tubes;  the  gas 
from  the  apparatus  is  received  in  eprouvettes  over  a  bowl  of  potash 
solution  to  catch  the  carbonic  acid.  While  the  furnace  is  in  activity, 
a  strongly  explosive  gaseous  mixture  is  obtained  composed  of  the 
elements  of  water,  oxygen,  and  hydrogen.  Thus  a  part  of  the 
water  vapor  is  spontaneously  decomposed  or  dissociated  in  the 
porous  earthenware  tube;  the  hydrogen,  following  the  ordinary 
laws  of  osmosis,  has  passed  through  the  semi-permeable  partition 
and  is  separated  as  if  by  the  action  of  a  simple  filter  from  the 
oxygen  which  stays  in  the  interior  tube;  furthermore,  there  is 
found  with  this  oxygen  a  considerable  quantity  of  carbonic  acid 
coming  from  outside. 

In  general,  when  water  vapor  traverses,  without  special  pre- 
caution, a  strongly  heated  tube,  only  water  vapor  is  collected 
at  the  outlet  and  not  oxygen  nor  hydrogen;  in  fact,  the  water 
vapor,  decomposed  in  the  hotter  parts  of  the  apparatus,  is  totally 
re-formed  in  the  cooler  parts  passed  through  by  the  gases  result- 
ing from  this  decomposition.  If,  however,  the  passage  of  the 
water  vapor  through  a  strongly  heated  tube  is  extremely  rapid, 
and  if  the  water  vapor  is  mixed  with  a  large  excess  of  carbonic 
acid  whose  presence  hinders  the  recombination  of  the  oxygen  and 
hydrogen,  a  small  quantity  of  explosive  gas  may  be  collected  at 
the  outlet,  and  the  dissociation  undergone  by  water  vapor  at  a 
high  temperature  may  be  shown  in  a  very  simple  manner. 


CHEMICAL  EQUILIBRIUM.  59 

51.  Dissociation  of  carbonic  acid  gas. — This  simple  apparatus 
serves  also  to  show  another  decomposition,  one  not  less  remark- 
able than  that  of  water  vapor:  the  decomposition  of  carbonic  acid 
gas  at  high  temperatures. 

It  is  sufficient  to  pass  a  current  of  quite  pure  carbonic  acid 
through  a  small-bore  porcelain  tube  filled  with  pieces  of  porcelain 
and  heated  in  a  combustion-furnace  to  the  highest  temperature 
possible  (1200°  to  1300°). 

The  gases  coming  from  the  tube  pass  through  long  tubes  filled 
with  potash  solution,  where  they  are  separated  from  the  excess  of 
carbonic  acid. 

The  carbonic  acid  is  decomposed  by  heat  into  carbonous  oxide 
and  oxygen,  and  if  these  gases  do  not  totally  recombine  on  reach- 
ing the  colder  parts  of  the  apparatus,  this  is  probably  due  to  the 
difficulty  with  which  their  mixture  ignites  when  disseminated 
through  a  large  mass  of  inert  gas,  as  carbonic  acid. 

52.  These  decompositions  are  not  complete  but  limited;    at 
the  temperatures  at  which  they  are  produced,  the  inverse  reac- 
tion also  takes  place. — Must  we  conclude  from  these  observations 
that  at  the  temperatures  attained  in  the  investigations  of  H.  Sainte- 
Claire  Deville  water  is  totally  decomposed  into  oxygen  and  hydro- 
gen, carbonic  acid  totally  broken  up  into  oxygen  and  carbonous 
oxide?    If  this  were  so,  we  should  encounter  this  incomprehen- 
sible paradox:  Water  does  not  exist  at  the  temperature  of  melting 
silver,  and  nevertheless  oxygen  and  hydrogen  in  combining  pro- 
duce a  temperature  such  that  their  flame  fuses  iridium;  how  is  it 
that  this  flame  melts  platinum  and  that  melted  platinum  decom- 
poses water? 

It  is  clear  that  the  decomposition  of  water  at  a  given  tempera- 
ture should  not  be  total,  but  partial:  this  decomposition  should 
stop  when  the  gaseous  mixture  formed  by  the  water  vapor  and  by 
the  oxygen  and  hydrogen  which  comes  from  its  decomposition 
has  a  certain  composition ;  this  composition,  for  which  the  system 
is  in  equilibrium,  would  naturally  depend  upon  the  temperature. 

Conversely,  when  a  mixture  of  hydrogen  and  oxygen  is  brought 
to  a  temperature  sufficient  to  ignite  it,  the  combination  of  the  two 
gases  should  not  be  complete;  it  should  cease  for  a  certain  pro- 
portion of  water  vapor,  variable  with  temperature;  this  is  the 


60  THERMODYNAMICS  AND  CHEMISTRY. 

conclusion  reached  by  H.  Sainte-Claire  Deville  from  the  analysis 
of  the  properties  of  the  oxyhydrogen  flame. 

If,  as  was  the  case  before  the  researches  of  H.  Sainte-Claire 
Deville,  we  accept  the  hypothesis  that  above  500°  oxygen  and 
hydrogen  combine  entirely  to  form  water  vapor,  it  is  easy  to  cal- 
culate the  temperature  reached  in  the  oxyhydrogen  flame.  The 
computation  requires  a  knowledge  only  of  the  specific  heat  of 
water  vapor  and  the  heat  of  formation  of  water.  In  this  way  is 
found  the  excessively  high  temperature  of  6800°. 

Now  this  temperature  seems  quite  improbable.  It  is  true 
that  the  oxyhydrogen  flame  melts  platinum-iridium,  but  its 
temperature  should  not  much  exceed  the  point  of  fusion  of  this 
alloy,  for  the  alloy  placed  in  the  flame  is  hardly  brighter  than  at 
its  fusing-point. 

This  temperature  may  even  be  determined  approximately: 
by  pouring  into  cold  water  considerable  masses  of  melted  platinum 
or  iridium  brought  to  the  highest  possible  temperature  given  by 
oxygen  and  hydrogen  which  combine  in  equal  equivalents,  and 
observing  the  maximum  rise  of  temperature  produced  in  the 
water,  it  is  found  that  the  fixed  point  of  the  combination  of  these 
two  gases  cannot  exceed  2500°,  and  is  perhaps  less. 

How  explain  these  results?  Evidently  they  are  due  to  the 
fact  that  in  the  oxyhydrogen  flame  the  gases  which  burn  do  not 
totally  combine;  a  part  of  these  gases  escapes  combustion. 

When  oxygen  and  hydrogen,  mixed  in  equal  equivalents,  burn 
there  is  formed  a  certain  quantity  of  water  vapor,  but  a  certain 
quantity  of  hydrogen  and  of  oxygen  remain  in  a  state  of  liberty. 
With  a  special  arrangement,  the  gases  which  feed  the  oxyhydrogen 
flame  may  be  drawn  partly  off,  and  it  is  found  that  the  hottest  parts 
always  contain  uncombined  oxygen  and  hydrogen. 

If  this  experiment  is  repeated  with  a  burner  of  oxygen  and 
carbonous  oxide,  it  is  readily  seen  that  the  flame  is  far  from 
uniquely  made  of  carbonic  acid ;  in  the  hottest  part  of  the  flame 
two  thirds  at  the  most  of  the  gases  oxygen  and  carbonous  oxide 
are  combined;  it  is  only  in  the  coolest  part  of  the  flame  that  the 
combustion  is  total. 

These  various  experiments  put  beyond  doubt  the  following 
propositions: 


CHEMICAL  EQUILIBRIUM.  61 

At  a  high  temperature  water  and  carbonic  acid  are  decomposed 
but  the  decomposition  is  not  complete;  an  equilibrium  is  set  up 
when  the  mixture  formed  by  the  compound  and  its  component 
gases  has  reached  a  certain  composition;  the  mixture  in  equilib- 
rium contains  a  less  proportion  of  the  compound  considered  in  the 
proportion  that  the  temperature  is  higher. 

Conversely,  at  a  high  temperature  oxygen  and  hydrogen  com- 
bine; carbonous  oxide  and  oxygen  form  carbonic  acid,  but  the 
combination  is  not  total;  it  tends  towards  a  state  of  equilibrium 
at  which  it  stops;  in  this  equilibrium  state  the  proportion  of  the 
gases  which  have  escaped  combination  is  the  greater  as  the  tem- 
perature is  higher. 

53.  Example  of  a  state  of  equilibrium  which  is  the  common 
limit  of  two  reactions  the  inverse  of  each  other.  Action  of  water 
vapor  on  iron  and  the  inverse  action. — The  preceding  experi- 
ments show  that  hi  the  same  system  which  includes  a  molecule  of 
oxygen  and  one  of  hydrogen,  at  the  same  temperature,  two  inverse 
reactions  may  be  observed :  decomposition  of  water  vapor  and  for- 
mation of  water  vapor;  they  show  us  that  each  of  the  two  inverse 
reactions  stops  when  the  system  has  reached  a  certain  state  of 
equilibrium;  but  they  do  not  show  us  that  these  two  states  of 
equilibrium  are  identical  with  each  other.  We  shall  see  farther  on, 
when  we  study  the  states  of  false  equilibrium  (Chap.  XVIII),  that 
it  is  not  useless  to  demonstrate  experimentally  this  equality. 

Here  is  a  case  in  which  experiment  shows  states  of  equilibrium 
of  which  each  is  the  common  limit  of  two  inverse  reactions,  and 
for  which  the  laws  governing  these  states  of  equilibrium  may  be 
completely  analyzed. 

At  a  high  temperature  iron  reduces  water  vapor,  giving  the 
magnetic  oxide  of  iron;  inversely, by  passing  a  current  of  hydrogen 
over  the  magnetic  ferrous  oxide,  iron  and  water  vapor  are  obtained; 
H.  Sainte-Claire  Deville l  and  Debray 2  have  tried  to  determine 
the  conditions  in  which  these  two  inverse  reactions  are  produced. 

A  porcelain  tube  containing  iron  and  the  magnetic  oxide  were 
immersed  in  a  bath  brought  to  a  fixed  temperature;  hydrogen 

1  H.  SAINTE-CLAIRE  DEVILLE,  Comptes  Rendus,  v.  70,  pp.  1189  and  1201, 
1870;  v.  71,  p.  30,  1871. 

2  H.  DEBRAY,  Comptes  Rendus,  v.  88,  p.  1341,  1879. 


62 


THERMODYNAMICS  AND   CHEMISTRY. 


could  be  admitted  to  this  tube,  which  also  received  water  vapor 
coming  from  a  flask  filled  with  cold  water;  in  accordance  with 
Watt's  principle,  the  tension  of  the  water  vapor  in  the  whole  appa- 
ratus was  equal  to  the  tension  of  the  saturated  water  vapor  at  the 
temperature  of  the  flask,  and  consequently  had  a  known  value; 
a  manometer  gave  the  pressure  of  the  mixture  of  hydrogen  and 
water  vapor  and  thus,  by  difference,  the  pressure  of  the  hydrogen. 

Suppose,  for  example,  the  flask  brought  to  the  temperature  at 
which  the  pressure  of  the  saturated  water  vapor  is  4.6  mm.,  and  heat 
the  porcelain  tube  to  200° ;  so  long  as  the  pressure  of  the  hydrogen 
is  less  than  95.7  mm.,  the  water  vapor  attacks  the  iron,  reaction 
having  for  effect  the  increase  of  the  hydrogen  pressure;  when,  on 
the  contrary,  this  pressure  becomes  greater  than  95.7  mm.,  it 
diminishes,  because  a  part  of  the  hydrogen  is  used  to  reduce  the 
ferrous  oxide ;  when  the  temperature  is  200°  and  the  vapor  pressure 
of  the  water  4.6  mm.,  the  system  is  in  a  state  of  equilibrium  corre- 
sponding to  the  value  95.7  mm.  for  the  pressure  of  hydrogen;  the 
system,  removed  from  this  state  in  the  one  or  the  other  direction, 
undergoes  a  chemical  reaction  which  brings  it  back;  this  state  is 
therefore  one  of  stable  equilibrium. 

This  state  varies  with  the  temperature;  the  vapor  pressure  of 
water  being  always  4.6  mm.,  the  hydrogen  pressure  at  the  instant 
of  equilibrium  has,  at  various  temperatures,  the  values  given  in 
the  following  table : 


Temperature. 

Hydrogen  Pressure. 

200° 

95  7  mm 

Mercury  boiling-point  

40  5    " 

Sulphur. 

25  8    " 

Cadmium         " 

12  9    " 

Zinc                  " 

92" 

About  1000°.  . 

51" 

At  a  given  temperature  this  state  of  equilibrium  changes  with 
-the  vapor  pressure  of  water;  the  hydrogen  pressure,  at  each  tem- 
perature, is  approximately  proportional  to  the  vapor  pressure  of 
the  water  vapor;  thus  at  200°  when  the  water  vapor  has  a  pressure 
of  4.6  mm.,  the  hydrogen  pressure  at  the  instant  of  equilibrium 
has  the  value  95.7  mm.,  whose  ratio  to  the  vapor  pressure  of  the 


CHEMICAL  EQUILIBRIUM.  63 

water  is  20-8;  at  the  same  temperature,  when  the  water-vapor  has  a 
pressure  of  9.7  mm.,  the  pressure  of  the  hydrogen,  at  the  instant 
of  equilibrium,  is  195  mm.,  giving  the  ratio  of  20.1. 

These  observations  show  us,  for  high  temperatures,  what  the 
phenomena  of  etherification  and  the  double  decomposition  of  salts 
showed  us  for  ordinary  temperatures:  namely,  the  existence,  in  a 
chemical  system,  of  a  state  of  equilibrium,  the  common  limit  of 
two  reactions  the  inverse  of  each  other. 

54.  Changes  of  physical  state  give  rise  to  equilibrium  con- 
ditions each  of  which  is  the  common  limit  of  two  modifications 
the   inverse   of   each   other.      Saturation  of  solutions. — Similar 
conclusions  hold  for  changes  of  physical  state  as  were  furnished 
by  chemical  reactions. 

Take,  at  0°,  an  aqueous  solution  of  sodium  chloride  in  the 
presence  of  a  crystal  of  this  salt.  If  the  solution  contains  less 
than  36  grammes  of  salt  to  100  grammes  of  water,  it  dissolves  the 
new  pieces  of  salt  until  the  concentration  corresponds  to  36  grammes 
of  salt  dissolved  in  100  of  water;  then  the  transformation  considered 
ceases  and  the  solution  is  saturated.  If  the  solution  contains  more 
than  36  grammes  of  salt  to  100  grammes  of  water,  salt  is  precipi- 
tated and  the  solution  attains,  without  exceeding  it,  the  concen- 
tration 36/100. 

A  system  formed  of  water  and  salt  is  then  in  equilibrium,  at 
the  temperature  0°,  when  the  solution  contains  36  grammes  of 
salt  dissolved  in  100  grammes  of  water.  This  state  of  equilibrium 
is  the  common  limit  of  two  inverse  modifications,  solution  and 
precipitation. 

55.  Another    example.      Tension    of     saturated    vapor. — 
Another  example  is  furnished  by  the  vaporization  of  water.     A 
vessel  containing  water  and  water  vapor  is  brought  to  100°  C. 
Whatever  may  be  the  form  and  size  of  the  vessel  and  the  respec- 
tive masses  of  water  and  vapor,  the  following  facts  are  observed: 

If  the  pressure  in  the  vessel  is  inferior  to  the  pressure  of  1 
atmosphere,  the  water  is  changed  into  vapor;  the  vaporization 
ceases  when  the  pressure  becomes  1  atmosphere. 

If  the  pressure  is  greater  than  one  atmosphere,  the  vapor  con- 
denses; the  condensation  ceases  at  atmospheric  pressure. 

At  100°,  liquid  water  and  water  vapor  are  in  equilibrium  if  the 


64 


THERMODYNAMICS  AND  CHEMISTRY. 


pressure  in  the  vessel  is  equal  to  the  pressure  of  1  atmosphere; 
this  state  of  equilibrium  is  the  common  limit  of  two  changes  of 
state  the  inverse  of  each  other,  vaporization  and  condensation. 
These  facts  are  included  in  the  well-known  law: 
At  a  given  temperature  a  liquid  of  definite  composition  is  in 
equilibrium  with  its  own  vapor  when  the  pressure  supported  by 
these  fluids  has  a  certain  value;  this  value  does  not  depend  upon 
the  size  or  the  form  of  the  containing  vessel,  of  the  masses  of  the 
liquid  and  vapor;  it  depends  solely  upon  the  nature  of  the  liquid 
and  upon  the  temperature;  it  is  called  the  tension  of  saturated 
vapor  of  the  given  fluid  at  the  temperature  considered. 

The  tension  of  saturated  vapor  of  a  given  fluid  increases  with 
the  temperature. 

At  a  given  temperature  the  liquid  vaporizes  if  the  pressure  is 
less  than  the  tension  of  saturated  vapor  for  the  temperature 
considered;  the  vapor  condenses,  on  the  contrary,  if  the  pressure 
is  greater  than  the  tension  of  saturated  vapor. 

This  law  is  susceptible  of  a  common  geometrical  representa- 
tion. 

Take  two  coordinate  axes  OT,  OP  (Fig.  19),  of  which  OT  is 
the  axis  of  temperatures  and  OP  of 
pressures. 

Let  F  be  the  tension  of  saturated 
vapor  of  a  certain  fluid  at  the  tem- 
perature t;  the  point  S  whose  abscissa 
is  t  and  whose  ordinate  is  F  shows  the 
conditions  in  which  the  liquid  will  be 
in  proper  equilibrium  with  its  own 
vapor. 

When  the  temperature  t  varies, 
the  tension  F  varies  also  and  the  point 
S  describes  a  certain  curved  line  VV 
which  is  the  vapor-pressure  curve  for 
saturated  vapor  of  the  fluid  considered. 


M 


FIG.  19. 


Since  the  tension  of  saturated  vapor  F  increases  at  the  same 
time  as  the  temperature  t  we  see  that  the  vapor-pressure  curve 
rises  from  left  to  right. 

At  the  temperature  t  take  a  pressure  P,  greater  than  the  ten- 


CHEMICAL  EQUILIBRIUM.  65 

sion  F  of  the  saturated  vapor;  the  point  M,  of  abscissa  t  and 
ordinate  P,  will  be  above  the  point  S  belonging  to  the  curve  VV'm 
Take  likewise  a  pressure  p,  less  than  the  tension  F  of  saturated 
vapor;  the  point  m,  of  abscissa  t  and  ordinate  p,  will  be  below 
the  point  S. 

The  law  stated  above  may  be  expressed  in  the  following  way: 

The  curve  of  tensions  for  a  saturated  vapor,  each  point  of 
which  represents  a  state  of  equilibrium  of  the  system  formed  by 
the  liquid  and  its  vapor,  is  the  boundary  of  two  regions. 

Every  point  in  the  region  situated  above  the  vapor-pressure 
curve  represents  a  state  of  the  system  in  which  the  liquid  vaporizes. 

Every  point  in  the  region  below  this  curve  represents  a  state  in 
which  the  vapor  condenses. 

56.  Dissociation  of  calcium  carbonate.  Tension  of  disso- 
ciation.— Guided  by  the  intuitions  of  H.  Sainte-Claire  Deville, 
H.  Debray  *  has  proved  that  the  laws  of  the  vaporization  of  a  liquid, 
laws  which  we  have  just  recalled,  may  be  applied  almost  literally 
to  the  chemical  decomposition  of  certain  bodies,  notably  to  the 
dissociation  of  carbonate  of  calcium  into  lime  and  carbonic  acid 
gas. 

Heat,  at  a  known  temperature  t,  carbonate  of  lime  in  a  vessel 
which  communicates  with  a  mercury  pump;  this  pump  serves 
either  to  remove  the  carbonic  acid  produced  or  to  force  in  carbonic 
acid,  and  at  the  same  time  gives  a  measure  of  the  gaseous  pressure 
at  each  instant. 

At  a  given  temperature  t  the  decomposition  of  the  carbonate 
of  lime  stops  when  the  pressure  of  the  carbonic  acid  reaches  a 
certain  value  F;  if,  keeping  the  temperature  constant,  the  car- 
bonic acid  produced  is  removed  by  the  pump,  a  new  decompo- 
sition takes  place,  which  again  stops  when  the  pressure  of  the 
carbonic  acid  retakes  the  value  F;  if  carbonic  acid  gas  is  forced 
into  the  apparatus,  this  gas  combines  with  the  free  lime  until  the 
pressure  returns  to  the  value  F.  These  results,  similar  to  those 
that  would  be  had  if  the  vessel  contained  a  solid  or  liquid  in  the 
presence  of  its  vapor,  may  be  expressed  by  saying  that  the  space 
containing  the  carbonate  of  lime  is  saturated  with  carbonic  acid 
gas  when  this  gas  is  at  the  pressure  F. 

1  H.  DEBRAY,  Comptes  Rendus,  v.  64,  p.  603,  1867. 


66 


THERMODYNAMICS  AND   CHEMISTRY. 


At  a  given  temperature  t,  this  pressure  has  a  value  which  does 
not  depend  upon  the  various  peculiarities  which  may  characterize 
the  experiment;  in  particular,  it  does  not  change  if  at  the  begin- 
ning of  the  experiment  there  is  put  into  the  apparatus  not  only 
Iceland  spar,  but  also  an  excess  of  quicklime.  Depending  solely 
upon  the  temperature  t,  this  pressure  may  be  called  the  dissocia- 
tion tension  of  calcium  carbonate  at  the  temperature  t. 

The  dissociation  tension  of  carbonate  of  calcium  at  a  tempera- 
ture t  varies  with  this  temperature  and  increases  with  it.  H. 
Debray  found  the  following  values  for  this  tension : 


Temperature. 

Dissociation  Tension. 

Mercury  boiling-point  
Sulphur                           

Zero 
Hardly  sensible 

Cadmium            "           
Zinc                     "           

85  mm. 
520    " 

The  decomposition  of  calcium  carbonate  is  not  the  only  reac- 
tion showing  a  dissociation  tension,  fixed  at  each  temperature 
and  similar  in  all  points  to  the  tension  of  saturated  vapors;  H. 
Debray  found  the  same  law  in  studying  *  the  decomposition  of  a 
certain  number  of  hydrated  salts  into  water  vapor  and  anhydrides. 

We  may  evidently  construct,  for  each  of  the  cases  of  which  we 
have  spoken,  a  curve  of  dissociation  tensions,  which  possesses  all  the 
properties  of  the  vapor-pressure  curve  for  saturated  vapors.  In 
each  of  the  reactions  studied  by  Debray  the  determinations  of 
this  investigator  give  us  a  certain  number  of  points  on  the  curve 
of  dissociation  tensions;  but  these  points  are  too  few  and  too 
widely  separated  from  each  other  to  allow  us  to  draw  the  curve. 

Isambert  has  tried  to  fill  this  gap.  He  made  use  of 2  the  com- 
pounds that  certain  metallic  chlorides,  bromides,  or  iodides  form 
with  ammonia  gas. 

At  a  given  temperature  there  is  emission  or  absorption  of 
ammonia  gas  according  as  the  pressure  of  this  gas  is  less  or 
greater  than  a  certain  dissociation  tension.  The  dissociation 
tension  at  a  given  temperature  depends  exclusively  upon  this 

1  H.  DEBRAY,  Comptes  Rendus,  v.  66,  p.  194,  1868. 

2  ISAMBERT,  Comptes  Rendus,  v.  66,  p.  1259,  1868;    Annales  de  I'Ecole 
Normak  Superieure,  v.  5,  p.  129,  1868. 


CHEMICAL  EQUILIBRIUM.  67 

temperature  and  upon  the  ammonia  compound  which  is  destroyed 
or  formed  under  pressures  near  to  this  tension.  Isambert  was 
able  to  determine  the  curves  of  dissociation  tension  for  a  certain 
number  of  ammonia  compounds. 

Since  the  time  when  Isambert  published  this  work  several 
chemists  have  made  known  a  great  number  of  dissociation  ten- 
sion curves.  These  curves  have  exactly  the  appearance  and  prop- 
erties of  those  for  the  tension  of  saturated  vapors  of  solids  or  liquids. 

57.  The  study  of  chemical  reactions  and  the  study  of  physi- 
cal changes  of  state  depend  on  the  same  theory,  that  of  chemical 
mechanics. — These  observations  clearly  show   that  chemical  re- 
actions and  physical    changes  of  state  sometimes  obey  exactly 
similar  laws;    consequently  every  theory  applicable    to   chemical 
reactions  in    general    should   include  also    physical  changes   of 
state. 

From  the  beginning  of  its  deve'opment,  thermodynamics  was 
applied  successfully  to  vaporization,  fusion,  and  solution.  It  is 
there  ore  natural  to  seek  to  extend  it  to  chemical  reactions. 

This  attempt,  crowned  with  success,  has  given  birth  to  chemi- 
cal mechanics  founded  upon  thermodynamics,  the  subject  of  these 
lessons. 

58.  Idea  of  reversible  transformation. — It  is    to    thermody- 
namics that  we  must  look  for  the  constitution  of  a  truly  rational 
chemical  mechanics.     We  shall  see  this  chemical  mechanics  de- 
veloped from  the  union  of  the  principle  of  the  equivalence  between 
work  and  heat  with  a  new  principle,  discovered  by  Sadi  Carnot, 
transformed  and  enlarged  by  Clausius. 

In  order  to  state  this  principle,  it  is  necessary  for  us  to  become 
acquainted  with  one  of  the  most  delicate  principles  in  all  thermo- 
dynamics, that  of  reversible  transformation. 

Take  a  system,  acted  upon  by  certain  forces,  in  a  given  state  1 
and  suppose  that,  under  the  action  of  these  forres.  the  system 
passes  to  the  state  2.  It  has  undergone  a  modification  in  the 
course  of  which  it  has  passed  through  various  states  which  succeed 
e^ch  other  in  a  continuous  manner.  As  soon  as  the  system,  during 
this  modification,  is  in  any  one  of  these  states,  it  immedatiely 
leaves  it  to  pass  into  the  state  following,  which  proves  this  state 
passed  through  by  the  system  not  to  have  been  one  of  equilibrium. 


68  THERMODYNAMICS  AND  CHEMISTRY. 

When  the  forces  which  act  on  the  system  and  the  conditions 
in  which  it  is  placed  are  given,  the  nature  and  the  direction  of  the 
modification  which  it  undergoes  are  necessarily  fixed.  If,  for 
example,  these  forces  and  these  conditions  cause  the  system  to  go 
from  the  state  1  to  the  state  2  passing  through  the  intermediate 
states  A.  B,  C,  D  .  .  . ,  it  cannot  be  admitted  that,  placed  in  the 
same  conditions  and  acted  upon  by  the  same  forces,  the  system  can 
repass  from  the  state  2  to  the  state  1  by  traversing  in  inverse  order 
precisely  the  same  states  .  .  .  D,  C,  B,  A;  this  is  expressed  by 
saying  that  a  real  transformation  is  never  reversible. 

It  is  not  reversible,  but  it  may  be  renversable;   it  will  be  ren- 
versable  if,  by  modifying  the  forces  which  act  upon  the  system  and 
the  conditions  m  which  it  is  placed,  the  system  may  be  made  to 
repass  from  the  state  2  to  the  state  1 ;  but  in  general,  in  this  trans- 
formation, the  inverse  of  the  preceding,  the  system  will  pass  through 
the  states  .  .  .  D',  C',  Bf,  A'  different  from  the  states  .  .  .  D, 
C,  B,  A,  either  in  the  properties  possessed  by  the  body  in  each  of 
these  states  or  in  the  forces  which  act  upon  it. 
Take  an  example  borrowed  from  mechanics. 
In  an  Atwood's  machine  (F  g.  20)  let  a  weight  n  pass  from  the 
x-x  level  A  to  the  level  Z  under  the  action  of  the 

counterweight  TT',  less  than  TT;  at  the  moment 
that  the  weight  TT  passes  the  level  N,  between 
A  and  Z,  its  velocity  of  fall  is  not  zero;  it  is 
not  therefore  in  equilibrium. 

_A  Take  the  weight  TT  at  the  level  Z;   give  to 

rli'          it  as  initial  velocity  the  velocity  directed  from 
11  J»  N       above  downwards  that  it  had  at  the  end  of  the 

preceding  fall;  under  the  action  of  the  same 
counterweight,  the  weight  TT  will  not  remount 


FIG.  20.  from  the  level  Z  to  the  level  A ;  it  will  con- 

tinue, on  the  contrary,  to  descend;    the  modification  considered 
is  not  reversible. 

It  is,  however,  renversable;  by  taking  a  counterweight  TT' 
greater  than  TT,  the  weight  n  may  be  brought  back  from  the  level 
Z  to  the  same  level  A ;  but  at  the  instant  during  this  ascension 
when  it  repasses  the  level  N,  it  will  be  no  longer  acted  upon  by 
the  force  which  acted  upon  it  when,  in  descending,  it  passed  through 


CHEMICAL  EQUILIBRIUM.  69 

the  same  level  and  its  velocity,  instead  of  being  directed  down- 
wards, will  be  directed  upwards. 

Take  two  states  1  and  2  of  a  system;  suppose  that  certain 
forces  can  cause  the  system  to  pass  from  state  1  to  the  state  2, 
passing  through  a  series  of  intermediate  states  A,  B,  C,  D  .  .  .  ; 
that  other  forces  bring  it  from  the  state  2  to  the  state  1,  passing 
through  other  states  .  .  .  D',  C',  B',  A'.  Between  the  states  1 
and  2  arrange  a  series  of  equilibrium  positions  a,  ft,  7,  d,  .  .  .  fol- 
lowing each  other  in  a  continuous  manner.  The  system,  placed 
in  any  one  of  these  positions,  wrould  remain  there  indefinitely. 
This  series  of  positions  of  equilibrium  cannot  then  be  passed  through 
by  the  system  either  in  one  direction  or  in  the  other;  it  does  not 
correspond  to  a  realizable  transformation  of  the  system. 

Take  the  transformation  ABCD  .  .  .  which  brings  the  system 
from  state  1  to  state  2.  Change  slowly  the  forces  which  act  upon 
the  system  during  this  transformation  so  that  they  approach 
gradually  the  forces  which  would  assure  the  system's  equilibrium 
in  each  of  the  positions  through  which  it  passes;  suppose  that, 
by  the  effect  of  this  operation,  the  state  that  we  denote  by  A 
tends  towards  the  equilibrium  position  a,  that  the  state  we  call 
B  tends  towards  the  equilibrium  position  /?....  We  shall 
have  thus  constructed  a  continuous  series  of  realizable  transfor- 
mations suitable  to  make  the  system  pass  from  state  1  to  state  2 ; 
and  this  series  of  transformations  will  have  for  limiting  form  the 
chain  of  equilibrium  positions  a,  ft,  f,d...,  which  is  not  a 
realizable  modification. 

Suppose,  for  example,  that  in  our  Atwood's  machine  we  cause 
to  approach  n  the  counterweight  it'  (nf>n),  under  whose  action 
the  weight  n  falls  from  A  to  Z;  the  velocity  of  this  weight  it,  at 
the  instant  it  passes  the  level  N,  will  approach  0 ;  the  force  acting 
upon  it  will  approach  0;  the  state  of  the  weight  it,  at  the  instant 
it  passes  through  N,  will  approach  a  limiting  state  in  which  it 
would  be  kept  in  equilibrium  at  the  level  N,  by  an  equal  counter- 
weight; the  actual  descents  which  we  studied,  but  taken  more  and 
more  slowly,  will  have  for  1  miting  form  a  series  of  equilibrium 
positions  of  the  weight  TT  under  the  action  of  an  equal  counter- 
weight, positions  succeeding  each  other  with  continuity  from  the 
level  A  to  the  level  Z;  this  series  of  equilibrium  positions  is  the 


70  THERMODYNAMICS  AND  CHEMISTRY. 

limit  of  a  series  of  falls;  but  it  does  not  constitute  a  fall ;  the  weight  n 
cannot  really  pass  by  this  series  of  states,  which  does  not  prevent  us 
from  turning  our  attention,  if  we  wish,  to  each  one  of  these  states. 

Similarly,  if  we  modify  gradually  the  forces  which  act  upon 
the  system  to  make  it  pass  from  the  state  2  to  the  state  1  passing 
through  the  states  .  .  .  Df,  C',  B',  A',  we  could  cause  the  real 
transformation  .  .  .  D',  Cr,  B'}  A'  to  vary  in  a  continuous  manner 
and  admit  for  limiting  form  the  series  of  limiting  states  of  equi- 
librium .  .  .  d,  Y,  /?,  a,  which  is  not  a  realizable  transformation. 

In  our  Atwood's  machine,  for  example,  we  might  lift  the  weight 
x  from  Z  to  A  under  the  action  of  the  counterweight  n',  greater 
than  TT;  by  varying  in  a  cont'nuous  manner  the  counterweight 
;r',we  may  modify  this  lifting  in  a  continuous  manner  also;  if  the 
counterweight  TT'  approaches  TT  as  a  limit,  the  ascent  of  the  weight 
n,  becoming  slower  and  slower,  will  have  for  limiting  form  a  series 
of  equilibrium  states  of  the  weight  TT,  extended  in  a  continuous 
manner  from  the  level  Z  to  the  level  A. 

This  series  of  equilibrium  states  a,  ft,  /-,  d  .  .  .  which  is  passed 
over  by  no  modification  of  the  system  is,  in  some  sort,  the  common 
boundary  of  the  real  transformations  that  bring  the  system  from 
the  state  1  to  the  state  2  and  of  the  real  transformations  that  bring 
the  system  from  state  2  to  state  1;  change,  however  slightly,  in 
a  certain  direction,  the  conditions  that  maintain  the  system  in 
equilibrium  in  each  of  these  states  and  the  system  will  undergo  a 
real  transformation  which  will  conduct  it  from  state  1  to  state  2; 
change,  however  slightly,  these  conditions  in  the  opposite  direc- 
tion, and  you  obtain  a  real  transformation  from  state  2  to  state  1 ; 
this  series  of  equilibrium  states  is  called  a  revers  ble  transformation. 

Thus  the  reversible  transformation  is  a  continuous  series  of 
equilibrium  states;  it  is  essentially  unrealizable;  but  we  may  give 
our  attention  to  these  equ  librium  states  successively  either  in 
the  order  from  state  1  to  state  2,  or  in  the  reverse  order;  this 
purely  intellectual  operation  is  denoted  by  these  words:  to  cause 
a  system  to  undergo  the  reversible  transformation  considered,  either 
in  the  direction  1-2,  or  in  the  reverse  direction. 

59-  Example  of  a  reversible  transformation  furnished  by  the 
vaporization  of  a  liquid. — This  idea  of  reversible  transformation 
is  of  great  importance  in  all  branches  of  thermodynamics;  one  can- 


CHEMICAL  EQUILIBRIUM.  71 

not  insist  too  much  upon  it;  let  us  illustrate  it  by  several  examples 
taken  from  physical  changes  of  state  and  changes  of  chemical  con- 
stitution. 

At  the  temperature  of  100°  the  vapor  tension  of  water  is  equal 
to  1  atmosphere;  at  200°  it  is  15  atmospheres. 

Take  a  mass  of  water,  entirely  liquid,  at  the  temperature  of 
100°  and  under  atmospheric  pressure;  this  will  be  the  state  1  of 
our  system;  next  take  the  same  mass  of  water,  entirely  vapor,  at 
200°  and  under  15  atmospheres  pressure;  this  will  be  the  state  2 
of  our  system. 

We  may,  by  a  real  transformation,  cause  the  system  to  pass  from 
the  state  1  to  the  state  2 ;  heat  it  so  as  to  raise  the  temperature 
gradually  from  100°  to  200°;  at  the  same  time  increase  the  pres- 
sure, but  so  slowly  that  at  each  instant  its  value  is  less  than  the 
tension  of  saturated  water  vapor  for  the  temperature  of  the  system 
at  the  given  instant;  under  these  conditions  the  liquid  water 
will  be  changed  into  vapor  until  it  has  all  passed  over  to  this 
latter  state. 

Similarly,  by  a  real  transformation,  we  may  cause  the  system 
to  pass  from  state  2  to  state  1 ;  lower  gradually  the  temperature 
from  200°  to  100°;  at  the  same  time  decrease  the  pressure,  but 
so  slowly  that  its  value,  at  each  instant,  is  greater  than  the  tension 
of  saturated  water  vapor  for  the  temperature  at  which  the  system 
is  at  the  same  instant;  in  these  conditions  the  water  vapor  will 
continue  to  condense  until  it  has  all  passed  over  into  the  liquid 
state. 

Change  gradually  the  conditions  in  which  the  first  transforma- 
tion is  produced,  so  that  the  pressure  at  each  instant  is  nearer  and 
nearer  the  tension  of  saturated  vapor  for  the  temperature  of  the 
system  at  this  same  instant.  Change  also  the  conditions  under 
which  the  second  transformation  is  produced,  so  that  the  pressure 
exceeds  less  and  less  the  tension  of  saturated  vapor  for  the  tem- 
perature of  the  system  at  this  same  instant.  Our  two  real  trans- 
formations, the  inverse  of  each  other,  will  approach  the  same 
limiting  form;  this  limiting  form  will  be  composed  of  a  series  of 
systems  in  which  the  water  will  be  partly  liquid  and  partly 
vapor;  from  one  system  to  the  following  the  temperature  in- 
creases, the  liquid  mass  decreases,  the  mass  of  vapor  increases; 


72  THERMODYNAMICS  AND  CHEMISTRY. 

each  of  these  systems  will  support  a  pressure  which  is  precisely 
the  tension  of  saturated  water  vapor  for  the  temperature  to  which 
the  system  is  brought,  so  that  each  of  these  systems  is  in  equi- 
librium; it  will  be  the  seat  neither  of  a  vaporization  nor  of  a 
condensation. 

According  as  one  passes  mentally  over  this  series  of  states  of 
equilibrium  in  a  certain  order  or  in  the  reverse  order,  one  will  pass 
by  a  reversible  transformation  from  the  state  1  to  the  state  2  or 
from  state  2  to  state  1. 

60.  Example  of  a  reversible  transformation  furnished  by  the 
dissociation  of  cupric  oxide. — The  vaporization  of  a  liquid  has 
just  furnished  us  with  an  example  of  a  reversible  transformation; 
the  dissociation  of  a  solid  compound,  such  as  the  carbonate  of 
calcium  or  of  cupric  oxide,1  will  give  us  a  second  example  quite 
analogous  to  the  preceding. 

Take  a  certain  mass  of  cupric  oxide  (CuO)  at  the  temperature 
ti  and  under  the  pressure  Plt  equal  to  the  pressure  of  dissocia- 
tion for  the  temperature  ^;  next,  take  at  the  temperature  t2 
and  pressure  P2,  equal  to  the  dissociation  tension  relative  to 
the  temperature  t2,  cuprous  oxide  (Cu2O)  and  the  oxygen  fur- 
nished by  the  dissociation  of  this  mass  of  cupric  oxide.  From 
the  state  1  to  the  state  2  we  may  pass  through  a  real  transforma- 
tion, by  varying  the  temperature  from  tt  to  L  and  constantly 
keeping  the  system  under  a  pressure  less  than  the  dissociation 
tension  of  the  cupric  oxide  at  the  actual  temperature  of  this  system. 
We  might  also  pass  from  the  state  2  to  the  state  1  through  a  real 
transformation  by  varying  the  temperature  from  t^  to  tv  and  in 
maintaining  the  system  constantly  at  a  pressure  greater  than  the 
dissociation  tension  of  the  cupric  oxide  at  the  actual  temperature 
of  this  system.  These  two  groups  of  inverse  transformations  will 
have  a  common  boundary;  this  boundary  will  be  composed  of 
a  series  of  systems  containing  cupric  oxide,  cuprous  oxide,  and 
oxygen,  at  a  temperature  t,  comprised  between  ^  and  tv  and  under 
a  pressure  P  equal  to  the  dissociation  pressure  at  the  temperature 
t  ;  all  these  systems  will  be  in  equilibrium,  and  this  series  of  states 
of  equilibrium  will  form  a  reversible  transformation  joining  the  state 
1  to  the  state  2. 

1  DEBRAY  and  JOANNIS,  Comptes  Rendus,  v.  99,  pp.  583  and  688,  1884. 


CHEMICAL  EQUILIBRIUM.  73 

61.  Example  of  a  reversible  transformation  furnished  by  the 
dissociation  of  water  vapor. — Here  is  another  example  of  the  re- 
versible transformation,  taken  from  the  phenomena  of  dissociation. 

A  vessel  of  constant  volume  contains  water  vapor  and  the 
mixture  of  hydrogen  and  oxygen  produced  by  the  decomposition 
of  this  vapor;  at  a  given  temperature  t  such  a  system  is  in  equi- 
librium when  the  ratio  of  the  mass  of  non-decomposed  water  vapor 
to  the  total  mass  of  the  system  has  a  value  x;  this  value  x  depends 
on  the  temperature  t  and  diminishes  when  the  temperature  rises. 
If,  in  the  vessel  brought  to  the  temperature  t,  the  proportion  of 
water  vapor  is  greater  than  x,  the  water  dissociates,  while  the 
oxygen  and  hydrogen  cannot  combine ;  if,  on  the  contrary,  the  pro- 
portion of  water  vapor  is  less  than  x,  the  oxygen  and  hydrogen 
combine  while  the  water  vapor  cannot  dissociate. 

Take  the  two  temperatures  ^=1200°  and  ^=1500°;  let  xlf  % 
be  the  values  of  x  which  correspond  to  these  temperatures;  x, 
will  be  greater  than  x2. 

From  a  state  1,  where  the  temperature  is  ^=1200°  and  where 
the  proportion  of  water  vapor  is  xlf  to  a  state  2,  where  the  tem- 
perature is  £2=1500°  and  where  the  proportion  of  water  vapor  is 
xv  we  may  pass  through  a  real  transformation,  hi  the  course  of 
which  the  temperature  will  constantly  rise  while  the  water  vapor 
will  dissociate  unceasingly;  it  will  be  necessary  for  this  that  at 
each  instant  the  proportion  of  water  vapor  in  the  system  is  greater 
than  the  value  of  x  which  corresponds  to  the  temperature  of  the 
system  at  the  same  instant. 

We  may  also  pass  through  a  real  transformation  from  the  state 
2  to  the  state  1,  in  the  course  of  which  the  temperature  will  fall 
constantly,  while  the  oxygen  and  hydrogen  will  combine;  it  will 
be  necessary  for  this  that  at  each  instant  the  proportion  of  water 
vapor  in  the  system  is  less  than  the  value  of  x  which  corresponds  to 
the  temperature  of  the  system  at  the  same  moment. 

These  two  groups  of  inverse  transformations  will  admit  a 
limiting  form  which  is  common  to  both;  this  limiting  form  will 
consist  of  a  series  of  systems  in  equilibrium;  in  each  of  them  the 
temperature  will  have  a  value  t,  included  between  ^  and  tv  and 
the  proportion  of  water  vapor  will  have  the  value  x  which  corre- 
sponds to  the  temperature  t;  this  series  of  systems  in  equilibrium 


74  THERMODYNAMICS  AND  CHEMISTRY. 

will  form  a  reversible  transformation  permitting  passage  from  state 
1  to  state  2  and  .nversely. 

From  now  on,  we  shall  consider  a  state  of  chemical  equilibrium 
in  the  light  of  its  ability  to  take  part  in  reversible  transforma- 
tions; its  essential  character  will  no  longer  consist  in  the  absence 
of  all  change;  it  will  consist  rather  in  the  separation  of  states 
which  are  the  seat  of  a  transformation  of  definite  direction  from 
those  states  which  are  the  seat  of  a  transformation  in  the  oppo- 
site direction ;  we  may  then  characterize  such  a  state  of  chemical 
equilibrium  as  one  where  two  reactions,  the  inverse  of  each  other, 
limit  each  other. 


CHAPTER  V 
THE    PRINCIPLES    OF    CHEMICAL    STATICS. 

62.  Sadi  Carnot's  principle.  Generalization  of  this  prin- 
ciple by  Clausius. — In  1824  Sadi  Carnot  *  published  a  short  work 
on  the  mechanical  effects  of  heat;  depending  on  the  one  hand 
upon  the  impossibility  of  perpetual  motion,  on  the  other  hand 
upon  the  principle,  then  accepted  without  question,  that  around 
a  closed  cycle  a  system  undergoes  losses  and  absorptions  of  heat 
which  exactly  compensate  each  other,  he  demonstrated  a  theorem 
of  the  greatest  importance  both  for  the  theory  of  heat  and  for  the 
applications  of  this  science  to  heat-engines. 

The  discovery  of  the  principle  of  the  equivalence  between  heat 
and  work,  by  annihilating  one  of  the  two  postulates  on  which 
Carnot's  demonstration  was  based,  seemed  as  if  it  would  crush 
his  theorem  by  a  single  blow.  Happily  this  was  not  so ;  Carnot's 
theorem  was  not  incompatible  with  the  new  principle;  by  com- 
bining the  principle  of  the  equivalence  between  heat  and  work  with 
a  postulate  analogous  to  the  impossibility  of  perpetual  motion, 
this  theorem  may  be  proved  in  a  manner  more  precise  than  that 
given  it  by  its  discoverer. 

It  was  Clausius  who  in  an  imperishable  memoir  2  reconciled 
Carnot's  theorem  with  the  principle  of  equivalence.  But  Clausius 
did  not  limit  himself  to  the  realization  of  this  work,  which 
alone  would  have  assured  him  the  admiration  of  physicists.  He 

1  SADI  CARNOT,  Reflexions  sur  la  puissance  motrice  du  feu  et  sur  les  ma- 
chines  propres  d  augmenter  cette  puissance;   Paris,    1824.     Reprinted   1872 
et  seq.;  also  in  English  in  several  forms. 

2  R.  CLAUSIUS,  Poggendorf's  Annalen,  Bd.  79,  pp.  368  and  500,  1850. 

75 


76  THERMODYNAMICS  AND  CHEMISTRY. 

generalized  and  transformed  Carnot's  *  theorem  so  as  to  make  of 
it  one  of  the  most  important  and  fruitful  principles  of  natural 
philosophy.  It  is  with  justice  that  the  name  of  Principle  of  Carnot 
and  Clausius  is  given  to  the  great  law  that  they  established. 

This  general  law,  which  we  shall  regard  as  one  of  the  PRIMARY 
HYPOTHESES  on  which  thermodynamics  is  based,  and  which  there- 
fore we  place  in  the  same  rank  as  the  Principle  of  the  equivalence 
between  heat  and  work,  may  be  stated  in  a  very  simple  manner. 

Let  us  first  define  what  Clausius  calls  the  value  of  transforma- 
tion of  a  modification. 

This  modification  may  be  isothermal;  this  means  that  the 
system  undergoing  a  change  has  the  same  temperature  at  all  points 
and  that  this  temperature  remains  constant  during  the  whole 
modification;  in  this  case,  if  Q  is  the  quantity  of  heat  set  free  by 
the  system  which  undergoes  this  change  and  if  T  is  the  constant 
value  of  the  absolute  temperature,  the  value  of  transformation  for 
the  change  is 


In  general,  even  if  the  temperature  is  supposed  to  keep  the  same 
value  at  all  points  of  the  system,  this  value  varies  from  one  instant 
to  the  next,  and  that  in  a  continuous  manner;  the  definition  of 
the  value  of  transformation  is  then  somewhat  more  complicated. 

Let  T0  be  the  initial  value  and  T  the  final  value  of  the  absolute 
temperature. 

Separate  the  total  change  into  n  partial  modifications,  M0, 
Mv  .  .  .  ,  Mn_!.  Let  T0,  Tlf  .  .  .  ,  Tn_!  be  the  values  of  the 
temperature  at  the  beginning  of  the  various  modifications.  Let 
Qo,  Qv  -  •  •  >  Qn-i  be  the  -  quantities  of  heat  set  free  by  the 
system  during  these  modifications.  Form  the  sum 


•+'+  , 

nr»  Tr  m*:  i      •    •    •      i 

•*•  o      ±  1 


Next  increase  without  limit  the  number  n  of  the  partial  modi- 
fications into  which  the  total  change  is  divided,  so  that  each  of 

1  R.  CLAUSIUS,  Poggendorf's  Annalen,  Bd.  93,  p.  481,  1854;    Bd.   116, 
p.  73,  1862;   Bd.  125,  p.  353,  1865. 


THE  PRINCIPLES  OF  CHEMICAL  STATICS.  77 

these  partial  modifications  tends  to  disappear.  Seek  the  limit 
approached  by  the  preceding  sum.  By  definition  this  limit  is  the 
value  of  transformation  of  the  total  modifications  considered: 


The  statement  of  the  CARNOT-CLAUSIUS  PRINCIPLE  is  then  the 
following  : 

Suppose  that  a  system  undergoes  a  series  of  modifications  forming 
a  closed  cycle.  If  all  the  modifications  which  compose  the  cycle  are 
reversible,  the  value  of  transformation  for  the  cycle  is  zero: 

(2)  .  =  0. 

//  all  the  modifications  which  compose  the  cycle  are  real,  or  if 
certain  of  them  are  real  and  others  reversible,  the  value  of  trans- 
formation for  the  cycle  is  positive: 

(3)  £>0. 

63.  The  absolute  temperature  is  always  positive.  —  These 
statements  cause  the  absolute  temperature  to  play  an  important 
role;  let  us  consider  this. 

We  have  seen  (Chap.  II,  Art.  24)  that  when  a  centigrade-scale 
thermometer  constructed  with  a  perfect  gas  indicated  the  tem- 

perature t,  the  absolute  temperature  had  the  value  T=  —  \-t,  a 

being  the  coefficient  of  expansion  of  a  perfect  gas.  We  have  seen 
also  that  if  the  same  mass  of  gas,  at  the  centigrade  temperatures 
t  and  t',  and  under  the  pressures  P  and  P',  occupied  the  volumes 
V  and  V,  the  following  equation  holds  [Chap.  Ill,  eq.  (6')]: 


PV    P'V 


T        T' 

Suppose,  for  instance,  that  the  absolute  temperature  T'  is  that 
which  corresponds  to  0°  C.,  that  is,  to  melting  ice;  we  shall  then 

have  J'=0,  T'=-;  let  P0  and  V0  be  the  pressure  and  volume  of 
this  mass  of  gas  at  this  temperature;  at  an  absolute  temperature 


78  THERMODYNAMICS  AND   CHEMISTRY. 

T  the  same  mass  of  gas,  at  the  same  value  F0,  would  exert  a 
force  P  and  we  should  have 

P=P0aT. 

Imagine  that  this  mass  of  gas  is  cooled  until  the  corresponding 
absolute  temperature  becomes  zero,  then  negative;  the  pressure  of 
this  gas  whose  volume  is  supposed  invariable  would  also  become 
zero,  then  negative,  which  is  incompatible  with  the  properties  of 
a  gas. 

This  contradiction  may  be  resolved  in  two  ways:  it  may  be 
considered  as  a  proof  that  it  is  impossible  to  sufficiently  cool  a 
body  so  that  the  absolute  temperature  of  this  body  becomes  zero 
or  negative;  it  may  also  be  taken  as  a  proof  that  no  body  can  re- 
main in  the  state  of  a  perfect  gas  when  the  temperature  becomes 
sufficiently  low. 

As,  in  fact,  the  gases  approaching  the  ideal  state  differ  more 
and  more  from  this  state  when  the  temperature  is  lowered,  the 
second  statement  may  be  easily  accepted  and  the  reasoning  that 
we  have  just  developed  cannot  suffice  to  establish  the  first  affirma- 
tion. 

It  is  to  other  considerations,  which  bear  upon  the  definitions 
of  absolute  temperature  and  which  we  cannot  explain  here,  that  we 
must  look  for  the  justification  of  the  follow'ng  proposition  : 

It  is  impossible  to  sufficiently  cool  any  system  whatever  so  that 
the  absolute  temperature  of  this  system  becomes  zero  or  negative. 

The  absolute  zyro  of  temperature  would  thus  appear  to  be  a 
lower  limit  of  cold;  the  most  energetic  refrigerating  methods  that 
can  be  imagined  will  allow  us  to  approach  nearer  and  nearer  this 
limit;  they  will  never  let  us  reach  it. 

64.  Property  of  a  real  isothermal  cycle.  —  These  remarks 
will  be  useful  to  us  on  several  occasions.  Let  us  consider  a  first 
application. 

Imagine  that  a  system  actually  describes  a  closed  cycle  and 
that  during  the  cycle  the  absolute  temperature  keeps  the  constant 
value  T7;  we  then  have  a  real  isothermal  cycle,  for  which  we  may 
write  the  equation 


THE  PRINCIPLES  OF  CHEMICAL  STATICS. 
The  inequality  (3)  may  then  be  written 


But,  from  what  has  just  been  said,  the  factor  —  is  essentially 
positive;  the  preceding  inequality  then  becomes 


When  a  system  is  put  through  a  real  isothermal  cycle  it  always 
sets  free  more  heat  than  it  absorbs. 

Similarly,  if  we  denote  by  W0  the  kinetic  energy  that  the  sys- 
tem possesses  at  the  instant  it  began  to  describe  the  cycle,  by  W± 
the  kinetic  energy  it  possessed  at  the  moment  it  completed  this 
cycle,  by  W  e  the  work  done  during  the  description  of  the  cycle,  we 
shall  have,  by  the  principle  of  the  equivalence  between  heat  and 
work  [Chap.  II,  eq.  (2')], 


The  preceding  inequality  gives,  therefore, 
(4) 


65.  Impossibility  of  perpetual  motion.  —  Several  conclusions 
may  be  drawn  from  this  inequality. 

Suppose,  in  the  first  place,  that  it  is  question  of  a  system  re- 
lieved from  all  external  action;  We  in  this  case  becomes  zero; 
the  inequality  (4)  becomes 


When  a  system  at  constant  temperature,  relieved  of  the  action  of 
external  forces,  undergoes  a  transformation  which  returns  it  to  its 
initial  state,  it  returns  hairing  a  less  kinetic  energy  than  at  the  start. 

Thus  a  simple  machine,  whose  temperature  cannot  vary,  and 
which  must  pass  periodically  through  the  same  state,  will  so  pass  with 
a  slower  motion  at  each  return;  this  is  the  impossibility  of  perpetual 
motion,  a  subject  of  engrossing  study  of  mechanics  toward  the 
close  of  the  eighteenth  century  and  at  the  beginning  of  the  nine- 
teenth. 


80  THERMODYNAMICS  AND  CHEMISTRY. 

66.  Continuously  acting  machines. — Let  us  now  imagine   a 
continuously  acting  machine,  that  is  to  say,  a  system  which  peri- 
odically repasses  through  the  same  state  and  whose  various  parts 
retake  periodically  the  same  velocity,  which  restores  to  the  system 
the  same  kinetic  energy.      If  the  machine  is  kept  at  a  constant 
temperature,  each  of  these  periods  will  constitute  an  isothermal 
cycle,  to  which  will  apply  also  the  equality  W^  W 0.   The  inequality 
(4)  will  then  become 

We>0. 

In  order  to  keep  a  constant  temperature  machine  in  continuous 
operation,  it  is  necessary  that  the  external  forces  applied  to  this  ma- 
chine do  positive  work  for  each  period. 

Such  a  machine  would  not  constitute  a  motor;  a  motor,  in  fact, 
is  a  machine  which  compels  the  external  forces  to  act  in  a  sense 
contrary  to  their  natural  tendency — which  obliges,  for  instance, 
a  weight  to  rise;  each  period  of  a  motor  in  continuous  action  must 
correspond  to  negative  work  of  the  external  forces;  it  is  thus  evi- 
dent that  as  long  as  isothermal  machines  only  are  at  our  disposal 
we  have  not  a  true  motor;  to  realize  a  motor  capable  of  perma- 
nent action,  it  is  necessary  to  cons' der  a  system  which,  during  each 
period,  undergoes  variations  of  temperature;  it  is  necessary  to 
construct  a  heat-engine. 

67.  Statement  of  the  Carnot-Clausius  principle  for  an  open 
reversible   transformation.     Entropy. — These  propositions  suffice 
to  give  a  clue  to  the  importance  of  the  principle  of  Carnot  and 
Clausius  in  the  theory  of  heat-engines;  but  the  study  of  the  motive 
power  of  heat  is  not  our  object;    so  let  us  return  to  the  principle 
itself  and  try  to  put  it  into  such  a  form  that  it  may  be  easily 
applied  to  the  problems  of  chemical  mechanics. 

In  the  form  that  we  have  stated  it,  this  principle  applies  to 
a  closed  cycle  only ;  this  is  a  troublesome  condition  for  its  applica- 
tion. The  principle  of  the  equivalence  of  work  and  heat,  in  the  first 
form  stated,  possessed  the  same  inconvenience  (Chap.  II,  Art  21); 
we  transformed  it  in  such  a  way  as  to  remove  this  inconvenience, 
and  it  is  this  transformation  which  introduced  into  our  reasonings 
the  quantity  called  internal  energy.  We  shall  transform  the  theorem 
of  Carnot  and  Clausius  in  an  analogous  manner,  and  this  trans- 


THE  PRINCIPLES  OF  CHEMICAL   STATICS.  81 

formation  will  introduce  into  our  reasonings  another  quantity, 
entropy,  which  will  not  be  of  less  importance  than  internal  energy. 

Let  us  consider  in  the  first  place,  quite  exclusively,  reversible 
transformations. 

The  principle  of  Carnot  and  Clausius  then  takes  the  following 
form  :  If  a  system  describes  a  reversible  closed  cycle,  the  transforma- 
tion value  s,  calculated  for  the  whole  cycle,  is  equal  to  0. 

This  stated,  suppose  that  a  system  passes  from  any  initial 
state  0  to  a  final  state  1  by  a  definite  series  M  of  reversible  trans- 
formations; the  transformation  value,  calculated  for  this  first 
modification,  has  the  value  e;  next  suppose  that  the  system 
returns,  by  a  series  JJL  of  reversible  transformations,  from  the  state 
1  to  the  state  0;  the  transformation  value  calculated  for  this 
second  modification  has  the  value  y  ;  the  totality  of  these  two  series 
of  transformations  M  and  fi  forms  a  reversible  cycle  for  which  the 
transformation  value  is  (e+  y)  ;  the  principle  of  Carnot  and  Clausius 
requires  that 

(5)  e+9=0. 

Now  imagine  that  the  system  passes  from  the  same  initial  state 
0  to  the  same  final  state  1  by  a  series  of  reversible  transformations 
M',  differing  from  the  series  M;  for  this  series  of  transformations 
the  transformation  value  will  be  e';  next  suppose  that  the  system 
returns  to  the  state  0  from  the  state  1  by  the  series  /*  of  reversible 
transformations,  for  which  the  transformation  value  is  TJ  ;  we  shall 
have  this  time 


But  the  two  equalities  obtained  require  that 


so  that  the  principle  of  Carnot  and  Clausius  involves  the  following 
proposition  : 

By  whatever  reversible  path  a  system  passes  from  a  given  initial 
state  to  a  given  final  state,  the  transformation  value  remains  constant. 

It  is  clear,  besides,  that  this  proposition  involves  in  its  turn 
the  exactness  of  the  principle  of  Carnot  and  Clausius  for  a  reversi- 
ble closed  cycle;  in  fact,  if  the  system  describes  a  reversible  cycle, 


82  THERMODYNAMICS  AND  CHEMISTRY. 

the  transformation  value  will  have,  for  this  cycle,  the  same  value 
as  for  every  other  reversible  modification  taking  the  system  in  the 
same  initial  state  and  returning  it  to  this;  consequently  it  has 
the  same  value  as  if  the  system  was  not  modified  at  all,  and  this 
value  is  visibly  0;  thus,  for  any  reversible  cycle  whatever,  £=0 
which  is  the  statement  of  Carnot  and  Clausius's  principle  for 
such  a  cycle. 

Thus,  so  long  as  reversible  cycles  alone  are  considered,  the  princi- 
ple of  Carnot  and  Clausius  may  be  regarded  as  the  exact  equiva- 
lent of  the  following  proposition : 

The  transformation  value  of  a  reversible  modification  depends 
on  the  initial  and  final  states  of  the  system,  but  not  on  the  reversible 
modification  chosen  by  which  the  system  passes  from  this  initial  to 
this  final  state. 

It  suffices  now  to  resume  a  reasoning  similar  to  that  which  gave 
us  the  notion  of  potential  (Chap.  I,  Art.  10)  in  order  to  transform 
the  preceding  statement  into  the  following: 

For  every  state  X  of  the  system  there  corresponds  a  quantity  Sx, 
such  that  for  every  reversible  modification  which  causes  the  system  to 
pass  from  the  state  0  to  the  state  1  the  transformation  value  is  given 
by  the  equality 

e  =  S0  +  S1. 

Clausius,  to  whom  is  due  these  considerations,  gave  to  the 
quantity  Sx  the  name  entropy  of  the  system  taken  in  the  state  X. 

By  means  of  this  nomenclature  equation  (5)  may  be  stated  in 
the  following  way: 

The  transformation  value  of  any  reversible  modification  whatever 
is  equal  to  the  decrease  in  the  entropy  of  the  system  brought  about  by 
this  modification. 

This  proposition  includes  as  a  particular  case  the  proposition 
relative  to  a  reversible  closed  cycle.  A  closed  cycle  is  in  fact  a 
transformation  in  which  the  final  state  1  is  identical  with  the 
initial  state  0;  Sl  is  equal  to  S2  and  equation  (5)  gives  equation  (2). 

68.  Entropy  of  a  perfect  gas. — As  an  example,  let  us  calculate 
the  variation  in  entropy  of  a  perfect  gas  of  mass  M  when  this  gas 
passes  from  a  state  in  which  its  volume  is  F0  and  its  absolute  tem- 
perature T0  to  a  state  in  which  its  volume  is  V  and  its  absolute 
temperature  T. 


THE  PRINCIPLES  OF  CHEMICAL  STATICS.  83 

To  do  this  we  may  bring  the  gas  from  the  initial  state  to  the  final 
state  by  whatever  reversible  process  we  please;  we  shall  choose  the 
following  : 

1°.  The  gas  passes  from  the  volume  V0  to  the  volume  V,  the 
temperature  keeping  the  constant  value  TQ. 

2°.  At  the  constant  volume  V  the  gas  passes  from  the  tem- 
perature 770  to  the  temperature  T. 

The  first  modification  is  isothermal;  according  to  equation  (1) 

its  transformation  value  is  „-,  Q  being  the  quantity  of  heat  set 

*  0 

free  by  the  gas  during  this  modification. 

It  is  easy  to  calculate  this  quantity  of  heat  Q. 

The  temperature  being  constant,  the  internal  energy  of  the  gas 
does  not  change  (Art.  25).  The  modification  being  reversible, 
the  gas  is  constantly  in  equilibrium  and  the  kinetic  energy  is  con- 
stantly zero.  Equation  (4)  of  Chap.  II  gives 

W 


Finally,  the  work  of  exterior  pressure  is  given  by  equation  (11) 
of  Chap.  I,  which  may  be  written 

TF,  =  2.325P0F0(log  70-log  V). 
The  first  modification  furnishes  thus  a  transformation  value: 

F0-log  F). 


From  equation  (13)  of  Chap.  II  we  have 

P0V0=MRoT0, 
and  the  preceding  quantity  becomes 

(6)  2.325(log  yo-log  7). 


Let  us  now  consider  the  second  modification. 

Separate  it,  as  was  done  in  Art.  62,  into  n  partial  heatings,  all 
accomplished  at  the  constant  volume  V,  these  heatings  causing 
the  system  to  pass  first  from  the  temperature  T0  to  the  tempera- 
ture Tv  then  from  the  temperature  Tl  to  the  temperature  T2, 
finally  from  the  temperature  77n_1  to  the  temperature  T. 


84 


THERMODYNAMICS  AND  CHEMISTRY. 


If  we  denote  by  c  the  specific  heat  at  constant  volume  of  the 
gas  considered,  these  various  heatings  set  free  respectively  the 
following  quantities  of  heat: 


Form  the  sum 


and  find  the  limit  approached  by  this  sum  when  the  number  n 
of  partial  heatings  is  indefinitely  increased,  by  making  each  of 
them  approach  zero. 

Evidently  it  is  sufficient  for  this  to  find  the  limit  of  the  sum: 


T2-T 

m        '    I 


~t 


T-Tn_t_ 


Let  us  take  (Fig.  21)  two  coordinates  axes  OT,  Oy.    Upon  the 

axis  of  abscissas  OT  lay  off  the 
values  of  the  absolute  tempera- 
ture T.  For  each  abscissa  let 
there  correspond  a  point  M  whose 


coordinate  y  is  equal  to  „. 


The 


T, 


FIG.  21. 


locus  of  the  point  M  is  a  certain 

curve  CC";  y  being  smaller  as 
'  T  is  greater,  this  curve  descends 
_from  left  to  right;  it  is  one  of 
T  those  curves  we  studied  in  Art.  7 

and  that  the  geometers  call  equi- 

lateral hyperbolas. 


T  —T 

The  quantity  —  ^  —  -  is  measured  by  the  rectangle 


The  sum 


rn 

J-  1 


rp 
-I  n-i 


^ 


THE  PRINCIPLES  OF  CHEMICAL  STATICS.  85 

is  measured  by  the  area  included  between  the  straight  line  T0T, 
the  two  ordinates  T^M^,  T0m,  and  the  broken  line  M  0m1M1m2Af2 


When  the  points  of  division  Tlt  T2,  .  .  .  ,  Tn_lf  marked  between 
T0  and  Tl  become  more  and  more  numerous  and  nearer  and  nearer 
together,  all  the  points  on  this  broken  line  approach  the  points  of 
the  arc  M^M  of  the  equilateral  hyperbola  CC".  The  limit  of  the 
sum  I  therefore  measures  the  area  included  between  the  straight 
line  T0T,  the  two  ordinates  T0M0  and  TM,  and  the  arc  MJM  of 
the  equilateral  hyperbola. 

Now  we  said  in  Art.  7  that  the  geometers  can  evaluate  such 
an  area,  which  has  the  value  (denoting  by  log  the  common  loga- 
rithm) 

2.325  log  ^-=2.325(log  T-log  T0). 

*  0 

Thus  our  second  modification  has  the  transformation  value 
(7)  2.325Mc(log  T0-log  T). 

Uniting  the  results  of  (6)  and  (7),  keeping  in  mind  equation  (3) 
we  may  state  the  following  : 

When  a  perfect  gas  of  mass  M  passes  from  a  state  in  which  it 
occupies  the  volume  F0  and  in  which  its  absolute  temperature  is 
T0  to  a  state  in  which  its  volume  is  V  and  its  temperature  T,  its 
entropy  passes  from  the  value  S0  to  the  value  S,  and  we  have 


(8) 


S=S0+2.325M0^1og  V+c  log  T-^log  F0-c  log 


69.  Statement  of  the  principle  of  Carnot  and  Clausius  for  a 
real,  open  transformation.  Compensated  and  non-compensated 
transformations.  —  Let  us  now  consider  a  real  transformation  M 
which  causes  the  system  to  pass  from  the  initial  state  0  to  the  final 
state  1;  what  can  we  say  concerning  the  transformation  value  e 
for  this  change? 

Consider  a  reversible  transformation  M'  which  returns  the 
system  from  the  state  1  to  the  state  0;  according  to  equation  (5), 
the  modification  M'  has  for  transformation  value  (Si—S0).  The 


86  THERMODYNAMICS  AND   CHEMISTRY. 

ensemble  formed  by  the  modification  M  followed  by  the  modifica- 
tion Mf  has,  therefore,  for  transformation  value 


But  this  ensemble  of  changes  brings  back  the  system  to  its 
initial  state;  it  is  a  closed  cycle  composed  in  part  of  real  trans- 
formations, in  part  of  reversible  transformations;  from  this, 
according  to  the  inequality  (3),  its  transformation  value  must  be 
positive.  If,  therefore,  we  denote  by  P  a  quantity  essentially 
positive,  we  shall  have 


or 

(9)  s  =  S0- 

This  equality  characterizes  any  realizable  modification  whatr 
ever  bringing  the  system  from  the  initial  state  0  to  the  final  state  1. 

This  quantity  P,  of  which  we  know  nothing  except  that  it 
is  positive,  has  received  from  Clausius  the  name  of  non-compen- 
sated transformation  relative  to  the  realizable  change  which  causes 
the  system  to  pass  from  the  state  0  to  the  state  1. 

We  see  that  the  non-compensated  transformation  relative  to  any 
realizable  modification  whatever  is  positive.  If  we  compare  equa- 
tion (5)  with  equation  (9),  we  see  that  a  reversible  modification  is 
accompanied  by  a  non-compensated  transformation  equal  to  zero. 

We  shall  give  to  the  quantity  (SQ—  SJ,  which  reduces  to  zero 
for  every  closed  cycle,  reversible  or  not,  the  name  of  compensated 
transformation  relative  to  the  modification  which  causes  the  system 
to  pass  from  the  state  0  to  the  state  1  . 

A  fundamental  property  distinguishes  the  compensated  trans- 
formation from  the  non-compensated  transformation.  In  order 
to  know  the  compensated  transformation  which  accompanies  a 
modification  undergone  by  a  system,  it  is  sufficient  to  know  the 
initial  and  final  states  of  the  system;  on  the  contrary,  to  know  a 
non-compensated  transformation,  it  is  necessary,  at  least  in  general, 
to  know  the  whole  series  of  intermediary  states  that  the  system 
has  passed  through  and  the  actions  to  which  it  was  submitted 
while  it  passed  through  them. 


THE  PRINCIPLES  OF  CHEMICAL  STATICS.  87 

70.  Principle  of  the  increase  in  entropy  of  an  isolated  system. 
—  The  laws  that  we  have  just  stated  lead  to  very  important  con- 
sequences. 

Imagine  that  a  material  system  be  absolutely  isolated  in  space; 
around  it  there  is  nothing,  neither  ponderable  matter  nor  ether; 
nothing  that  may  furnish  it  with  heat  nor  take  away  heat. 

This  system  undergoes  a  certain  modification  which  brings  it 
from  the  state  0  to  the  state  1  ;  what,  for  this  modification,  is  the 
transformation  value?  As  was  done  in  Art.  63,  we  may  decom- 
pose this  modification  into  a  certain  number  of  partial  modifica- 
tions ra,  ra',  ra"  .  .  .  ;  we  shall  denote  by  q,  qf,  q"  .  .  .  the  quan- 
tities of  heat  set  free  by  the  system  in  these  various  modifications 
and  by  T,  Tr,  T"  .  .  .  the  values  of  the  absolute  temperature  at 
the  beginning  of  each  of  them;  we  shall  form  the  sum 


and  we  shall  find  the  limit  approached  by  this  sum  when  the 
changes  m,  mf,  m"  .  .  .  become  more  and  more  numerous. 

Now,  the  system  being  able  neither  to  gain  nor  to  lose  heat, 
each  of  the  quantities  q,  q',  q"  .  .  .  is  equal  to  zero;  it  is  the  same 
for  the  preceding  sum  constantly  and  therefore  for  its  limit. 

Hence,  when  an  isolated  system  undergoes  a  real  modification, 
the  transformation  value  for  this  modification  is  zero  : 

£  =  0. 

This  equality  taken  with  equation  (9)  gives 

St-St-P, 

or,  since  P  is  essentially  positive, 


The  modification  considered  thus  causes  the  entropy  of  the 
isolated  system  to  increase. 

On  the  other  hand,  we  have  seen  (Chap.  II,  Art.  23)  that  such 
a  modification  left  the  total  energy  of  the  system  unchanged.  We 
are  therefore  led  to  the  following  statement,  which  was  given  by 
Clausius:  l 

1  R.  CLAUSIUS,  Poggendorf's  Annakn,  Bd.  125,  p.  400,  1865. 


88  THERMODYNAMICS  AND  CHEMISTRY. 

If  we  consider  a  system  absolutely  isolated  in  space,  every  real 
modification  of  this  system  has  the  following  properties: 
1°.  It  leaves  its  total  energy  constant. 
2°.  It  causes  the  entropy  to  increase. 

71.  Use  of  this  principle  in  chemical  statics. — The  proposition 
that  we  have  just  established  is  of  use  in  chemical  statics. 

The  chemical  system  A  is  in  a  certain  state  and  is  submitted 
to  the  action  of  other  bodies  B',  it  is  proposed  to  determine  if,  in 
these  conditions,  the  system  A  remains  in  equilibrium. 

The  system  A  and  all  the  bodies  B  which  act  upon  it  may 
compose  a  system  isolated  in  space;  let  C  be  the  total  system; 
we  shall  find  the  entropy  S  of  the  system  C. 

We  then  consider  all  the  modifications  by  which  the  system 
A  may  be  imagined  to  change  from  the  given  state,  and  we  calculate 
the  change  in  value  that  they  would  give  to  the  entropy  S  of  the 
total  system  C.  If  none  of  these  modifications  causes  the  entropy 
S  to  increase,  none  of  them  can  be  realized  and  the  system  A  re- 
mains forcibly  in  equilibrium  in  the  state  considered. 

Such  is  the  principle  of  chemical  statics  proposed  by  Horst- 
mann  1  and  by  J.  Willard  Gibbs.2  In  its  first  form  it  has  to  be 
used  somewhat  carefully  on  account  of  the  obligation  to  reckon 
in  the  calculation  of  the  entropy,  not  only  with  the  system  A  that 
is  studied,  but  also  with  all  the  foreign  bodies  B  which  act  on  it. 

We  shall  substitute  another  principle,  theoretically  equiva- 
lent, but  more  convenient  to  use. 

72.  Compensated  and  non-compensated  work  in  an  isother- 
mal modification. — Let  us  turn  to  equation  (9)  and  see  what  it 
becomes  for  an  isothermal  change. 

If  Q  is  the  quantity  of  heat  set  free  in  this  change  taking  place 
at  the  absolute  temperature  T7,  the  transformation  value  is,  accord- 
to  equation  (1), 


1  HORSTMANN,  Tkeorie  der  Dissociation  (Liebig's  Annakn  der  Chimie  und 
Pharmacie,  Bd.  170,  p.  192,  1873). 

2  J.  WILLARD  GIBBS,  On  the  equilibrium  of  heterogeneous  substances  (Trans- 
actions of  the  Acad.  of  Sci.  and  Arts  of  Conn.,  v.  3,  1875  to  1878). 


THE  PRINCIPLES  OF  CHEMICAL  STATICS.  89 

so  that  equation  (9)  becomes,  for  an  isothermal  modification 


or 

(10)  Q 

The  first  member  of  this  equation  (10)  being  a  quantity  of 
heat,  the  two  terms  which  compose  the  second  member  must  be 
quantities  of  the  same  kind  as  a  quantity  of  heat;  it  is  natural  to 
call  the  first,  T(S0—S1),  the  quantity  of  compensated  heat  set  free 
by  the  isothermal  change,  and  the  second,  TP,  the  quantity  of 
non-compensated  heat  set  free  by  the  same  change. 

The  first,  the  quantity  of  compensated  heat,  has  a  definite 
value  when  we  know  the  initial  state  0  and  the  final  state  1 
between  which  the  modification  is  taken.  This  is  not  so  for  the 
second;  like  the  quantity  P,  it  cannot  be  determined  if  the  initial 
and  final  states  alone  are  known,  but  only  when  all  the  intermediate 
states  through  which  the  system  has  passed  and  the  external  forces 
which  have  obliged  the  system  to  pass  from  each  to  the  following 
are  known. 

These  two  quantities  of  heat  are  distinguished  by  a  still  more 
essential  property. 

The  sign  of  the  quantity  of  compensated  heat  has  nothing 
arbitrary  about  it;  if  an  isothermal  change  which  brings  the  system 
from  the  state  0  to  the  state  1  liberates  a  positive  quantity  of 
compensated  heat,  a  change  leading  from  the  state  1  to  the  state 
0  will  liberate  a  negative  quantity  of  heat  equal  in  absolute 
value  to  the  preceding;  in  other  words,  it  will  absorb  as  much 
compensated  heat  as  the  first  liberated. 

On  the  contrary,  the  absolute  temperature  T  is  always  positive; 
the  non-compensated  transformation  P,  equal  to  0  for  a  reversible 
transformation,  is  positive  for  every  realizable  transformation; 
therefore  the  quantity  of  non-compensated  heat  liberated  by  any 
realizable  isothermal  change  is  a  quantity  essentially  positive;  it  is 
equal  to  0  for  a  reversible  change. 


90  THERMODYNAMICS  AND  CHEMISTRY. 

Multiply  the  two  members  of  equation  (10)  by  the  mechanical 
equivalent  of  heat  E]  this  equality  becomes 

(11)  EQ  =  ET(S9-Sd+ETP. 

The  first  member  of  this  equation,  product  of  a  quantity  of 
»  heat  by  the  mechanical  equivalent  of  heat,  is  a  quantity  of  the 
same  kincj  as  work;   it  is  therefore  the  same  for  each  of  the  two 
terms  composing  the  second  member.     We  shall  give  to  the  first, 

(12)  $  =  ET(S0-Sl), 

the  name  of   compensated  work  accomplished  in  the  isothermal 
change  considered,  and  to  the  second, 

(13)  r=ETP, 

the  name  of  uncompensated  work. 

Each  of  these  two  quantities  of  work  possesses  naturally  the 
same  characteristics  as  the  quantity  of  heat  to  which  it  is  equiva- 
lent. In  particular  we  may  state  this  fundamental  proposition: 

Every  real  isothermal  change  engenders  positive  non-compensated 
work;  this  work  is  zero  for  a  reversible  isothermal  modification. 

73.  First  form  of  the  equilibrium  condition  of  a  system  kept 
at  a  given  temperature. — This  proposition  will  furnish  us  with  a 
criterion  by  which  we  may  recognize  if  a  system  is  in  equilibrium 
at  a  given  temperature  in  a  given  state. 

Suppose  that  we  examine  all  the  infinitely  small  isothermal 
modifications  which  would  be  susceptible,  if  they  were  realized, 
of  making  the  system  leave  its  actual  state;  calculate  the  non- 
compensated  work  that  each  of  these  virtual  modifications  would 
cause;  if  all  these  modifications  correspond  to  a  zero  or  negative 
non-compensated  work,  none  of  them  is  realizable;  the  system 
cannot  quit  the  state  in  which  it  is;  it  remains  there  forcibly  in 
equilibrium;  hence  the  following  theorem: 

A  system  taken  in  a  given  state,  at  a  given  temperature,  is  in 
equilibrium  if  all  the  isothermal  modifications  by  which  it  may  be 
conceived  to  quit  this  state  correspond  to  a  zero  or  negative  non-com- 
pensated work. 


THE  PRINCIPLES  OF  CHEMICAL  STATICS.  91 

74.  Expression  for  non-compensated  work  accomplished  in 
an  isothermal  modification.  —  Let  us  compare  the  equations  (11) 
and  (13);  we  find 

r=EQ-ET(S0-Si). 

But  if  we  denote  by  W  the  kinetic  energy  of  the  system  in  a 
given  state,  by  U  the  internal  energy  in  this  same  state,  by  We 
the  work  done  by  the  external  forces  during  the  modification 
considered,  the  principle  of  the  equivalence  of  heat  and  work 
gives  [Chap.  II,  eq.  (4)] 


These  two  equations  give  us  the  following  expression  for  the 
non-compensated  work  produced  by  a  system  which  undergoes 
an  isothermal  modification: 

(14)  r^EWi-TSJ-EWi-TSJ  +  W.+Wt-Wv 

This  equation  (14)  gives  rise  to  a  great  number  of  important 
remarks. 

75.  Characteristic  function  of  a  system.  —  In  the  first  place 
equation  (14)  shows  that,  in  order  to  form  the  expression  for  non- 
compensated  work  engendered  in  an  isothermal  change,  it  is  not 
necessary  to  seek  separately  the  variation  of  internal  energy  and 
the  variation  of  entropy;  it  suffices  to  calculate  the  variation 
undergone  by  a  certain  quantity,  entirely  determined  when  the 
state  of  the  system  is  known;  it  is  the  quantity 

(15)  ff  =  E(U-TS). 

In  thermodynamics  this  quantity  plays  a  fundamental  role. 
Every  equation  furnished  by  thermodynamics  is,  in  the  last  analysis, 
the  statement  of  the  quantity  £F.  When  the  value  that  this  quan- 
tity takes  in  each  state  of  the  system  is  known,  one  may,  by  ordi- 
nary mathematical  operations,  determine  the  value,  in  each  state, 
of  the  internal  energy  and  entropy  of  the  system,  the  external 
forces  that  must  be  applied  to  keep  it  in  equilibrium  in  this  state, 
the  quantity  of  heat  that  must  be  furnished  to  raise  its  tempera- 


92  THERMODYNAMICS  AND  CHEMISTRY. 

ture  or  to  produce  any  modification;  it  is  possible,  in  a  word,  to 
determine  all  the  mechanical  and  thermal  properties  of  the  system. 
The  knowledge  of  the  value  that  the  quantity  ^  assumes  for  each 
state  of  the  system  therefore  characterizes  this  system.  Hence 
this  quantity  has  been  called  the  characteristic  function  of  the  system 
by  Massieu,1  who  first  treated  it  and  established  its  principal 
properties. 

76.  Characteristic  function  of  a  perfect  gas.  —  Let  us  calcu- 
late, as  an  example,  the  characteristic  function  of  a  perfect  gas 
of  mass  M  occupying  the  volume  V  at  the  temperature  T.  Let 
us  choose  arbitrarily  an  initial  state  in  which  the  gas  occupies  the 
volume  F0  at  the  temperature  T0;  let  U0,  S0  be  the  internal  energy 
and  entropy  of  the  gas  in  this  state;  let  U,  S  be  the  values  of  the 
same  quantities  when  the  gas  occupies  the  volume  V  at  the  tem- 
perature T.  Equation  (9)  of  Chap.  II  and  equation  (8)  of  the 
present  chapter  give 

U=UQ-McTQ+McT, 

~]og  70+clog 

+2.325M^|-  log  7+c  log  T). 

Denote  by 

A=E(U,-McT0), 

~  log  F0+c  log  !T0   -ESQ, 


two  quantities  which  do  not  depend  upon  V  and  T,  and  which 
therefore  remain  constant  as  V  and  T  vary;  hence  equation  (15) 
will  give  the  following  expression  for  the  characteristic  function 
of  a  perfect  gas: 

(16)     $  =  A+BT+EMcT(l-2.325  log  T)-2325MRaTlog  V. 

1  F.  MASSIEU,  Sur  les  fonctions  caracteristiques,  Comptes  Rendus,  v.  69, 
pp.  858  and  1057,  1869;  Memoire  sur  les  fonctions  caracteristiques,  Savants 
Strangers,  v.  22,  1876. 


THE  PRINCIPLES  OF  CHEMICAL  STATICS.  93 

77.  The  characteristic  function  considered  as  available  energy. 

— Equations  (14)  and  (15)  give  us  the  following  equality,  appli- 
cable to  any  isothermal  change: 

(17)  ^-TFo-TF.-SFo-JF.-T. 

Let  us  see  what  this  equation  teaches. 

The  change  considered  may  be  used  to  produce  work,  that  is, 
to  constrain  the  external  forces  to  do  negative  work;  We  being 
this  negative  work  of  the  external  forces,  the  work  produced,  W/, 
is  the  absolute  value  of  this  negative  work:  We'=—W9.  This 
change  may  also  be  used  to  increase  the  kinetic  energy  of  a  part  of 
the  system,  as  in  the  bow  when  the  spring  of  the  cord  sends  forth 
the  arrow.  In  a  general  way,  if  the  change  considered  is  used  as 
source  of  motive  power,  it  may  be  said  that  its  useful  effect  is 
measured  by  the  quantity 

(18)  S-TI^-TFo-TP.. 

In  virtue  of  this  equation,  we  may  write  equation  (17)  as 

(19)  a-fro-^-T. 

If  we  remember  that  the  non-compensated  work  r,  equal  to 
zero  for  every  reversible  isothermal  change,  is  positive  for  every 
realizable  isothermal  change,  we  obtain  without  difficulty  the 
following  proposition: 

//  we  consider  all  the  isothermal  modifications  susceptible  of  causing 
a  system  to  pass  from  a  given  initial  state  0  to  a  final  state  1,  also 
known,  it  may  be  shown  that  the  useful  effect  of  these  changes  is 
always  less  than  the  decrease  (SF0  — SFj)  in  the  quantity  &  when  the 
system  passes  from  the  state  0  to  the  state  1.  The  useful  effect  ap- 
proaches this  higher  limit,  without  reaching  it,  when  the  realizable 
isothermal  change  approaches  a  reversible  change. 

This  fundamental  proposition  seems  to  have  been  appreciated 
for  the  first  time  by  Maxwell;1  in  the  first  editions  of  his  book 
Maxwell  gave  to  the  quantity  £F  the  name  entropy  of  the  system, 
already  employed  by  Clausius  in  a  different  sense;  in  the  fourth 

1  J.  CLERK  MAXWELL,  Theory  of  Heat,  p.  186  (London,  1871). 


94  THERMODYNAMICS  AND  CHEMISTRY. 

edition  he  calls  it  available  energy,  which  had  been  suggested  by 
Gibbs;  *  Helmholtz  2  gave  it  the  name  of  free  energy;  to-day  the 
name  ordinarily  usel  is  the  internal  thermodynamic  potential  of 
the  system;  this  name  results  from  the  role  that  this  quantity 
plays  in  the  study  of  equilibrium  conditions  in  a  system. 

78.  Definite  form  of  the  equilibrium  condition  of  a  system 
kept  at  a  given  constant  temperature. — Referring  to  equation  (17), 
we  may  write  it  in  the  form 

(170  ^0-^1  +  ^=  r  +  W,-WQ. 

Suppose  the  system  placed  in  the  state  0  without  any  initial 
velocities  being  imparted  to  its  various  parts;  will  it  remain  in 
equilibrium  in  this  state  or  will  it  undergo  some  isothermal  change, 
any  reaction  that  will  disturb  this  equilibrium? 

Imagine  that  an  isothermal  reaction  brings  it  from  the  state  0 
to  the  state  1,  near  together;  the  initial  kinetic  energy  was  0; 
the  kinetic  energy  at  the  instant  when  the  system  is  in  the 
state  1  can  be  only  positive  or  zero,  and  similarly  for  (Wl  —  WQ); 
as  to  the  non-compensated  work  r,  it  can  only  be  positive;  there- 
fore every  isothermal  modification  capable  of  removing  the  sys- 
tem from  the  state  0,  where  it  was  without  initial  kinetic  energy, 
corresponds  to  a  positive  value  of  the  quantity  (i:+Wl  —  W0)  and 
consequently,  by  equation  (170,  of  the  quantity  (&Q  —  $i  +  We). 

Whence  the  following  proposition: 

A  system  placed  with  no  initial  velocity  in  the  state  0  remains 
of  necessity  there  in  equilibrium  if  all  the  virtual  isothermal  modifi- 
cations by  which  it  might  be  brought  from  the  state  0  to  a  neighboring 
state  1  verify  the  conditions 

(20)  ^0-^,  +  We<0. 

79.  Internal  thermodynamic  potential. — Referring    to    what 
was  said  in    Art.  14,  we  see  that  the  equilibrium  condition  just 


1  J.  WILLARD  GIBBS,  A  method  of  geometrical  representation  of  the  thermo- 
dynamic properties  of  substances  by  means  of  surfaces,  Trans.  Conn.  Acad., 
v.  2,  p.  400,  1873. 

2  H.  VON  HELMHOLTZ,  Zur  Thermodynamik  chemischer  Vorgdnge,  Sitzungs- 
berichte  der  Berliner  Akademie,  p.  2,  1882. 


THE  PRINCIPLES  OF  CHEMICAL  STATICS.  95 

stated  has  exactly  the  form  that,  in  mechanics,  the  equilibrium 
condition  of  the  system  constituted  as  follows  would  have: 

This  system  is  acted  upon  by  the  same  external  forces  as  the 
preceding;  it  is,  besides,  the  seat  of  internal  forces  whose  poten- 
tial is  SF  (Art.  10). 

Effectively,  in  a  modification  of  such  a  system,  while  the  ex- 
ternal forces  would  perform  work  equal  to  We,  the  internal  forces 
would  do  the  work  (&Q  —  &I)',  equation  (20)  would  therefore  ex- 
press the  fact  that  the  work  of  all  the  forces  acting  upon  the  system 
is  negative  or  zero  for  every  virtual  modification  capable  of 
removing  the  system  from  the  state  0.  We  come  therefore  to  the 
conclusion  : 

There  is  an  absolute  analogy  between  the  laws  of  equilibrium  in 
thermodynamics  and  the  statics  of  a  mechanical  system  in  which 
the  internal  forces  admit  a  potential;  the  same  role  is  plhyed  in  the 
latter  theory  by  the  potential  of  the  internal  forces,  and  in  the  former 
by  the  internal  thermodynamic  potential. 

So.  Total  thermodynamic  potential  under  constant  pressure  or 
at  constant  volume. — In  general  the  external  forces  acting  upon 
a  system  do  not  admit  a  potential;  but  there  are  always  special 
cases  which  admit  a  potential;  in  Art.  12  we  cited  two  that  are  of 
especial  interest  for  the  chemist: 

1°.  When  the  external  forces  reduce,  to  a  constant,  uniform 
pressure  n  they  admit  a  potential  of  value 

(21)  Q  =  nV. 

denoting  by  V  the  total  volume  of  the  system. 

2°.  When  the  external  forces  reduce  to  a  uniform  pressure, 
constant  or  variable,  if  the  total  volume  of  the  system  remains  fixed, 
the  external  forces  have  the  potential 

(22)  £=0. 

Let  us  consider  one  of  those  particular  cases  in  which  the  ex- 
ternal forces  applied  to  the  system  have  a  potential,  and  let  Q  be  this 
potential;  when  the  system  passes  from  the  state  0  to  the  state  1 
the  external  forces  will  do  work  given  by  the  equation 

(23)  We=Q.-Q\ 


96  THERMODYNAMICS  AND  CHEMISTRY. 

and  the  first  member  of  the  condition  (20)  becomes 


It  is  then  equal  to  the  increase  undergone,  from  the  state  0  to 
the  state  1,  by  a  certain  quantity 

(24)  &  =  $  +  £ 

which  has,  for  each  state  of  the  system,  a  definite  value.  This 
quantity,  the  sum  of  the  internal  thermodynamic  potential  and  of 
the  potential  of  the  external  forces,  is  called  total  thermodynamic 
potential. 

If  we  have  to  deal  with  a  system  kept  at  a  constant  uniform 
pressure  TT,  this  quantity  becomes,  by  equations  (21)  and  (24), 

(25)  0  =  ^+7:7. 

This  is  the  total  thermodynamic  potential  at  constant  pressure. 
If  the  system  is  kept  at  constant  volume  and  at  uniform  pres- 
sure, by  equations  (23)  and  (24)  this  quantity  becomes 

(26)  0=£F. 

This  is  the  total  thermodynamic  potential  at  constant  volume. 

For  the  case  in  which  the  system  studied  admits  a  total 
thermodynamic  potential  the  condition  (20)  becomes,  by  equa- 
tions (23)  and  (24), 

(27)  0o-0i^0. 

Placed  without  initial  kinetic  energy  in  a  state  0,  a  system  which 
admits  a  total  thermodynamic  potential  remains  in  equilibrium  in 
this  state  if  none  of  the  virtual  isothermal  modifications  which  may 
displace  it  do  not  cause  the  total  thermodynamic  potential  to  decrease. 

Hence  a  system  is  surely  in  equilibrium  in  a  state  in  which  the 
total  thermodynamic  potential  has  a  minimum  value  among  those  that 
it  may  assume  at  the  same  temperature. 

81.  Stability  of  equilibrium.  —  If  to  this  proposition  we  add 
the  complement  that  we  cannot  demonstrate  here:  Such  a  state 
is  one  of  stable  equilibrium,  we  shall  have  finished  noting  the  com- 
plete analogy  between  the  thermodynamic  statics  of  a  system 


THE  PRINCIPLES  OF  CHEMICAL  STATICS.  97 

which  admits  a  total  thermodynamic  potential  and  the  statics 
of  a  mechanical  system  acted  upon  by  forces  admitting  a  potential 
(see  Art.  16). 

82.  Interpretation  of  non-compensated  work.  —  Consider  again 
the  equation 

(10)  Q=T(S9-SJ  +  TP, 

from  which  we  have  already  drawn  so  many  conclusions.  We 
shall  find  an  interesting  interpretation  of  the  non-compensated 
heat  TP  set  free  in  this  modification. 

Let  us  suppose  the  kinetic  energy  of  the  system  constantly 
negligible,  which  is  ordinarily  true  for  the  changes  of  state  that 
the  chemist  studies;  equation  (17),  in  which  now  W0=Q,  PF,=0, 
gives 


or  from  equation  (13),  which  defines  r, 

(28)  W^S.- 


The  real  isothermal  change  considered  is  composed  of  a  series 
of  states  of  the  system;  imagine  that  by  means  of  external  actions 
suitably  chosen,  whose  nature  it  is  not  necessary  to  specify,  we 
may  keep  the  system  in  equilibrium  in  each  of  these  states.  The 
series  of  these  states  of  equilibrium  would  form  a  reversible  iso- 
thermal modification  bringing  the  system  from  the  same  initial 
state  0  to  the  same  final  state  1  as  the  real  modification  con- 
sidered; for  this  reversible  modification  the  non-compensated 
transformation  P  would  have  the  value  0;  if,  therefore,  we  de- 
note by  TF/  the  work  done  by  the  external  forces  which  act  upon 
the  system  during  this  reversible  modification,  equation  (28)  would 
give,  for  this  change, 

(280  HY-^-fro- 

Equations  (28)  and  (28')  give  the  equation 
(29)  ETP=We-W/. 

The  non-compensated  work  which  accompanies  a  real  isothermal 
modification  is  the  excess  of  the  work  done  during  this  modification 


98  THERMODYNAMICS  AND  CHEMISTRY. 

by  the  external  forces  really  acting  upon  the  system  over  the  work  that 
would  be  done  during  the  same  change  by  the  external  forces  capable 
of  maintaining  the  system  constantly  in  equilibrium. 

83.  Intensity  of  reaction;  slow  reactions.  —  We  often  speak 
in  chemistry  of  vigorous,  intense,  or  energetic  reactions  without 
giving  these  words  an  exact  meaning;  what  precedes  allows  us 
to  fill  this  gap;  we  may  take  the  difference  (We—  We'),  which  is 
always  positive,  as  a  measure  of  the  intensity  of  the  isothermal 
reaction  considered;  when  the  reaction  is  produced  under  condi- 
tions very  near  to  those  which  would  assure  equilibrium  at  every 
instant,  this  intensity  has  a  very  small  value;  to  have  a  large 
value,  it  is  necessary  that  the  reaction  be  produced  under  condi- 
tions extremely  different  from  those  assuring  equilibrium,  that 
we  have  a  system  very  much  out  of  equilibrium. 

By  means  of  equation  (29),  equation  (10)  becomes 

W  —W  ' 
(30)  Q 


Let  us  suppose,  in  the  first  place,  that  the  reaction  be  not  at 
all  intense;  the  quantity  (We  —  W  J}  will  have  a  very  small  value; 
the  sign  of  the  quantity  Q  will  be  that  of  T(S0—Sl);  but  this  last 
sign  is  not  arbitrary;  the  quantity  T(S0—S1)  may  be  either  posi- 
tive or  negative. 

Suppose,  for  instance,  that  for  this  change  the  quantity 
T(S0  —  Sl)  and  consequently  the  quantity  Q  are  positive;  the  iso- 
thermal change  considered  liberates  heat.  This  very  slightly  intense 
modification  is  very  nearly  reversible;  that  is  to  say,  in  condi- 
tions very  slightly  different  from  those  that  determine  the  modi- 
fications considered,  there  is  produced  a  modification  in  the  oppo- 
site direction,  absorbing  about  as  much  heat  as  the  first  sets  free. 

Hence  a  physical  or  chemical  change  of  very  slight  intensity 
may  either  absorb  or  liberate  heat. 

For  example,  accomplished  at  a  constant  temperature,  the 
vaporization  of  water  absorbs  heat  and  the  condensation  of  water 
vapor  liberates  heat. 

The  dissociation  of  the  carbonate  of  calcium  into  lime  and 
carbonic  acid  gas  absorbs  heat,  and  the  combination  of  carbonic 
acid  gas  with  lime  sets  free  heat. 


THE  PRINCIPLES  OF  CHEMICAL  STATICS.  99 

The  oxidation  of  •  iron  by  water  vapor  liberates  heat;  the  re- 
duction of  the  magnetic  oxide  by  hydrogen  absorbs  heat. 

84.  Very  intense  reactions;  principle  of  maximum  work. — 
Let  us  now  consider  the  case,  the  opposite  of  the  preceding,  in 
which  the  modification  studied  is  of  great  intensity;  the  difference 
(We—Wef),  which  is  always  positive,  has  a  very  great  value;  it 
therefore  gives  its  sign  to  the  quantity  Q.  Whence  the  following 
proposition : 

Every  isothermal  change  of  state,  of  great  intensity,  is  accom- 
panied by  a  liberation  of  heat. 

In  1854  J.  Thomsen1  stated  the  following  proposition,  which 
Berthelot  has  called  the  PRINCIPLE  OF  MAXIMUM  WORK: 

In  order  that  a  chemical  reaction  may  be  produced  at  a  tempera- 
ture kept  constant,  it  is  necessary  that  this  reaction  be  accompanied 
by  a  liberation  of  heat, 

Thomsen  took  care  to  limit  the  application  of  this  law  to  purely 
chemical  reactions;  it  is  but  too  clear  in  fact  that  it  may  not  be 
applied  to  physical  changes  of  state;  at  a  temperature  kept  con- 
stant, one  may  observe  the  vaporization  of  a  liquid,  the  fusion 
of  a  solid,  the  solution  of  sea-salt  in  water,  yet  all  these  changes 
of  state  absorb  heat. 

The  principle  of  maximum  work  may  not  even  be  regarded 
as  a  principle  applicable  to  all  chemical  reactions;  at  a  fixed  tem- 
perature, caibonate  of  calcium  dissociates,  hydrogen  reduces 
magnetic  iron  oxide,  in  spite  of  the  fact  that  these  reactions  absorb 
heat;  we  might  cite  an  immense  number  of  exceptions  to  the 
principle  of  maximum  work,  all  chosen  from  among  reactions  of 
feeble  intensity. 

The  principle  of  maximum  work  should  therefore  be  limited 
to  reactions  of  great  intensity,  for  which  it  is  applicable  from  the 
preceding  deductions. 

Thus  restricted,  this  principle  may,  in  a  great  number  of  cases, 
indicate  to  the  chemist  the  direction  of  possible  isothermal  re- 
actions. 

Let  us  consider  some  examples : 

The  combination  of  hydrogen  with  chlorine,  forming  a  mole- 

1  J.  THOMSEN,  Poggendorf's  Annalen  der  Physik  und  Chemie,  Bd.  92,  n.  34, 
1854. 


100  THERMODYNAMICS  AND   CHEMISTRY. 

cule  of  hydrochloric  acid,  liberates  22000  calories;  the  formation 
of  a  molecule  of  sulphurous  acid  gas  from  sulphur  and  oxygen 
sets  free  69260  calories;  also,  there  is  observed  the  direct  combi- 
nation of  hydrogen  and  chlorine,  of  sulphur  and  oxygen. 

Chlorine  and  oxygen,  in  forming  a  molecule  of  hypochlorous 
acid  gas,  would  absorb  15100  calories;  three  atoms  of  oxygen,  in 
uniting  to  form  one  molecule  of  ozone,  would  absorb  30700  calories; 
oxygen,  in  combining  with  water  to  form  a  molecule  of  hydrogen 
peroxide  with  a  great  excess  of  water,  would  give  rise  to  an  ab- 
sorption of  21700  calories;  also,  hypochlorous  acid  gas,  ozone, 
hydrogen  peroxide  are  substances  whi  h  decompose  sponta- 
neously. 

The  reaction 

Ag+Cl  =  AgCl 

sets  free  29000  calories,  while  the  reaction 

Ag  +  Br=AgBr 

sets  free  only  27100  calories.-  It  follows  from  this  that  chlorine 
will  displace  bromine  from  its  combination  with  silver,  for  the 
reaction 

AgBr  +€1 = AgCl + Br 

liberates  29000-27100  =  1900  calories. 
The  reaction 

2AgNO3  (dilute)  +Cu =Cu(N03)2(dilute)  +2Ag 
sets  free  25300  calories;  the  reaction 

Cu(NO3)2  (dilute)  +  Zn  =  Zn(N03)2(dilute)  +Cu 

liberates  61700  calories.  It  follows  that  copper  should  precipi- 
tate silver  from  a  dilute  solution  of  silver  nitrate,  and  zinc  should 
precipitate  copper  from  a  dilute  solution  of  copper  nitrate.  Ex- 
periment confirms  these  conclusions. 

A  compound  formed  with  absorption  of  heat  cannot,  accord- 
ing to  the  principle  of  maximum  work,  be  formed  in  a  direct  and 
isolated  manner;  but  an  endo thermic  compound  may  be  formed 
if  its  formation  is  the  necessary  consequence  of  reactions  whose 
ensemble  corresponds  to  a  liberation  of  heat. 


THE  PRINCIPLES  OF  CHEMICAL  STATICS.  101 

Thus  hydrogen  peroxide,  which  is,  as  we  have  just  seen,  an 
endo  thermic  compound,  may  be  formed  when  dilute  hydrochlo- 
ric acid  acts  on  barium  dioxide,  for  under  these  conditions  the 
reaction 

Ba02  +  2HC1  =  BaCl2  +  H202 

sets  free  22000  calories. 

These  examples,  which  might  be  indefinitely  multiplied,  will 
show  the  variety  of  cases  to  which  the  principle  stated  by 
Thomsen  may  be  applied.  We  must  not,  however,  entertain  any 
illusions  as  to  the  generality  of  this  principle. 

85.  Available  mechanical  effect  of  an  adiabatic  change.  — 
All  the  propositions  stated  from  Art.  72  to  the  preceding  (with 
the  exception  of  Arts.  75  and  76)  suppose  that  the  changes  im- 
posed upon  the  system  studied  are  isothermal,  that  is  to  say,  ac- 
complished without  any  change  of  temperature.  Without  this 
condition  these  propositions  may  lead  to  false  results.  We  shall 
give  an  example. 

Instead  of  treating  an  isothermal  change,  let  us  consider  an 
adiabatic  change;  such  a  change  is  so  called  in  thermodynamics 
when  the  system  neither  loses  nor  gains  heat. 

The  equation  which  expresses  the  principle  of  equivalence 
between  heat  and  work  [Chap.  II,  equation  (4)], 


becomes  in  the  present  case,  for  which  0=0, 
(31)  Wt-Wt-W 


Now  the  first  member  of  this  equation  is  what  we  have  called 
the  available  mechanical  effect  of  the  change  considered;  hence 
we  have  this  proposition: 

The  available  mechanical  effect  of  an  adiabatic  change  is  obtained 
by  multiplying  the  mechanical  equivalent  of  heat  by  the  decrease  sus- 
tained in  the  internal  energy  of  the  system  during  this  change. 

This  rule,  evidently,  is  very  different  from  the  one  obtained 
for  an  isothermal  change  (Art.  77). 

Suppose  that  the  adiabatic  change  considered  causes  the  sys- 
tem to  pass  from  the  state  0  to  the  state  1  in  which  it  has  the  same 


102  THERMODYNAMICS  AND  CHEMISTRY. 

temperature  as  in  the  state  0;  we  may  then  pass  by  an  isothermal 
change  from  the  same  state  0  to  the  same  state  1;  let  &Q  be  the 
available  mechanical  effect  of  the  adiabatic  modification  and  &r 
the  available  mechanical  effect  for  the  isothermal  change;  let  us 
compare  the  two  effects. 
Equation  (31)  gives 

(310  ^Q  =  E(UQ-Ul)f 

while  equation  (19)  gives 

Sr^o-SF-r. 

Taking  account  of  the  equation 
(15)  $  =  E(U-TS), 

it  is    lear  that  we  have 


But  by  equation  (11),  denoting  by  Q  the   quantity  of  heat  set 
free  in  the  isothermal  change,  this  equality  becomes 

(32)  &Q-&T  =  EQ. 

If  a  system  may  be  brought  from  the  same  initial  to  the  same  final 
state,  on  the  one  hand  by  an  isothermal  change,  on  the  other  hand 
by  an  adiabatic  change,  the  available  mechanical  effect  produced  in 
the  latter  case  exceeds  the  available  mechanical  effect  produced  in  the 
former  by  a  quantity  exactly  equal  to  the  quantity  of  heat  liberated 
by  the  isothermal  change. 

According  as  the  isothermal  reaction  is  exothermic  (Q>0)  or 
endothermic  (Q<0),  its  available  mechanical  effect  is  smaller 
(&T<&Q)  or  greater  (&T>&Q)  than  the  available  mechanical  effect 
of  the  Corresponding  adiabatic  change. 

Furthermore,  the  available  mechanical  effect  &Q  of  the  adia- 
batic change  may  exceed  the  limit  (&o-$J  that  the  mechanical 
effect  &T  of  the  isothermal  change  could  not  reach.  We  have 
in  fact 

SQ-OFo-frOK^o-tfiJ-^o-friX 
or,  by  equation  (15), 


THE  PRINCIPLES  OF  CHEMICAL  STATICS.  103 

The  mechanical  effect  of  an  adiabatic  change  for  which  the  final 
temperature  is  identical  with  the  initial  temperature  exceeds  the  limit 
(1F0— SFJ,  which  could  not  be  reached  by  the  mechanical  effect  of  an 
isothermal  change,  by  a  quantity  equal  to  the  compensated  work  which 
accompanies  this  latter  change. 

This  compensated  work  may  also  be  positive. 

86.  Application  to  the  theory  of  explosives. — These  considera- 
tions are  of  great  importance  in  the  theory  of  explosives. 

The  charge  which  detonates  in  pushing  the  ball  in  the  bore  of 
a  cannon  undergoes  such  rapid  reactions  that  we  may  regard  as 
quite  feeble  the  exchanges  of  heat  between  this  change  and  the 
metal  of  the  cannon  and  projectile.  For  a  first  approximation 
we  may  neglect  these  exchanges  and  treat  as  adiabatic  the  changes 
which  are  produced  in  the  bore  of  the  piece. 

The  charge  is  taken  at  the  ordinary  temperature,  which  we 
shall  call  T0;  at  the  moment  it  explodes,  the  temperature  of  the 
products  formed  increases  enormously;  then  these  products  ex- 
pand, expelling  the  projectile,  and  in  this  expansion  they  are 
cooled;  the  temperature  and  pressure  of  these  products  at  the 
moment  when  the  base  of  the  projectile  leaves  the  mouth  of  the 
piece  depend  evidently  upon  the  peculiarities  of  the  combustion 
of  the  powder  and  of  the  path  of  the  passage  of  the  bullet 
through  the  bore  of  the  piece.  Denote  by  1  this  final  state.  The 
mechanical  effect  produced  is 

(310  «e-tf(tf.-ffi), 

UQ  being  the  internal  energy  of  the  charge  in  the  initial  state  and 
(/!  the  internal  energy  of  the  products  of  the  combustion  taken 
in  the  final  state. 

Suppose  that  at  the  moment  the  projectile  quits  the  bore  of 
the  piece  the  combustion  of  the  powder  is  complete;  the  products 
of  the  combustion  at  this  moment  will  be  entirely  in  the  gaseous 
state,  and  we  may  without  serious  error  consider  them  as  perfect 
gases.  Then  to  know  the  internal  energy  U  it  will  be  sufficient 
to  know  the  mass  M  of  the  gaseous  mixture,  equal  to  the  mass 
of  the  charge  the  chemical  nature  of  this  mixture,  and  the  tem- 
perature Tl  to  which  it  is  brought;  it  will  not  be  necessary  to  know 
the  pressure  (see  Art  25). 


104  THERMODYNAMICS  AND  CHEMISTRY. 

The  temperature  Tt  differs  from  the  external  temperature  T0; 
in  all  firearms  it  is  notably  higher: 

(33)  T.-T.X). 

Let  UQ  be  the  internal  energy,  at  the  temperature  TQ,  of  the 
gaseous  mixture  furnished  by  the  combustion  of  the  explosive  in 
the  bore  of  the  piece;  let  c  be  the  specific  heat  at  constant  volume 
of  this  mixture;  we  shall  have  [Chap.  II,  eq.  (9)], 


Equation  (31')  will  then  become 

(34)  &Q  =  E(U. 

Taken  together  with  equation  (33),  this  equality  shows  that 
the  mechanical  effect  of  the  explosion,  supposed  adiabatic,  is  less 
than  the  quantity 

(35)  P=E(U0-u0). 

It  is  the  nearer  to  this  higher  limit  as  the  temperature  of  the 
gases  in  the  bore  of  the  piece,  at  the  moment  the  ball  quits  it,  is 
nearer  to  the  ordinary  temperature. 

This  quantity  P  represents  the  mechanical  effect  that  the 
charge  of  explosive  considered  would  produce  in  detonating  under 
the  most  favorable  conditions;  in  ballistics  it  is  called  the  potential 
of  this  explosive  charge. 

This  quantity  may  be  determined  by  means  of  data  furnished 
by  chemical  calorimetry. 

Suppose,  for  instance,  that  the  given  explosive  charge,  taken 
at  the  ordinary  temperature  T0  (or  +  15°  C.),  undergoes  at  constant 
volume  a  series  of  modifications  having  for  last  stage  a  gaseous  mix- 
ture of  the  same  composition  as  th  >  mixture  contained  in  the  piece 
at  the  departure  of  the  ball;  suppose,  besides,  that  at  the  end 
of  this  modification  the  temperature  becomes  T0.  This  change 
liberates  a  quantity  of  heat  1  The  kinetic  energy  is  zero  at  the 
beginning  and  at  the  end  of  the  change.  The  volume  of  the  sys- 
tem studied  has  remained  constant,  so  that  the  external  forces 
have  done  no  work.  The  principle  of  the  equivalence  of  work 
and  heat  gives,  therefore, 


THE  PRINCIPLES  OF  CHEMICAL  STATICS.  105 

or,  by  equation  (35), 

(36)  P=EL 

The  computation  of  the  explosive  potential  of  the  charge  is 
thus  reduced  to  the  determination  of  X,  that  is  to  say,  to  a  problem 
of  chemical  calorimetry.  The  whole  difficulty  of  the  problem, 
from  the  experimental  point  of  view,  consists  in  assuring  oneself 
that  the  reaction  produced  in  the  calorimeter  and  the  reaction 
produced  in  the  bore  of  the  piece  really  give  rise  to  the  same  gaseous 
mixture. 

Below  are  the  potentials  of  some  common  explosives  expressed 
in  gramme-metres.1 

POTENTIALS  OF  SOME  COMMON  EXPLOSIVES. 


No. 

Name. 

Potential. 

1 

War-powder  (black  powder)                 .    .          ... 

270  XlO3 

2 

Fulminate  of  mercury     .  .             

173     ' 

3 

Ammonium  nitrate  

267     ' 

4 

Picric  acid.  .  .             

323     ' 

5 

254     ' 

6 

457     ' 

7 

Octonitric  coton  (collodion)    .  . 

313     ' 

g 

Nitroglycerine                                                            .    . 

669     ' 

It  is  evident  that  it  is  useful  to  be  informed  in  an  exact  manner 
on  the  intensity  of  the  mechanical  effect  that  may  be  attained 
with  a  given  explosive  under  the  most  favorable  conditions. 

1  SARRAU,  Poudres  de  guerre,  ballistique  interieur,  Cours  de  T^cole  d'ap- 
plication  de  rartillerie  et  du  ge*nie,  1893. 


CHAPTER  VI. 
THE  PHASE  RULE. 

87.  The  number  of  independent  components  of  a  chemical 
system  of  given  kind. — After  having  discussed  the  principles 
of  thermodynamics  which  are  the  basis  of  modern  chemical  me- 
chanics, we  come  to  the  second  part  of  our  task:  the  exposition 
of  the  consequences  that  may  be  deduced  from  these  principles. 
It  is  naturally  by  the  aid  of  algebra  that  this  passage  from  princi- 
ples to  consequences  is  made;  but,  setting  aside  in  this  work  the 
mathematical  developments  that  the  reader,  if  he  desires,  may 
get  elsewhere,  we  shall  give  merely  the  results  and  compare  them 
with  the  teachings  of  experiment. 

When  we  have  reduced  the  representing  of  a  problem  by  an 
equation  to  be  no  more  than  an  algebraic  expression,  the  first  point 
we  have  to  examine  is  the  following:  How  many  distinct  quanti- 
ties are  there  in  the  equations  of  the  problem?  And  the  exami- 
nation of  this  point  is  immediately  followed  by  the  study  of  this 
other  point:  Among  these  distinct  quantities,  how  many  inde- 
pendent relations  does  algebra  furnish?  In  making  this  double 
enumeration  for  the  problems  of  chemical  mechanics,  J.  Willard 
Gibbs  was  led  to  the  propositions  whose  ensemble  constitutes  the 
phase  rule. 

Two  numbers  characterize  a  system:  the  number  of  inde- 
pendent components  which  form  it  and  the  number  of  phases  into 
which  it  is  divided. 

A  chemical  system  is  always  composed  of  a  certain  number  of 
simple  substances;  but  ordinarily,  by  reason  of  conditions  im- 
posed upon  the  system  studied,  conditions  that  give  in  a  sense 
the  definition  of  this  system,  determining  the  kind,  the  masses  of 

106 

\ 


THE  PHASE  RULE.  107 

these  various  elements  cannot  be  taken  arbitrarily;  there  exist 
certain  relations  among  them.  Thus,  a  system  containing  cal- 
cium carbonate,  lime,  and  carbonic  acid  gas  consists  of  calcium, 
carbon,  and  oxygen;  but  the  masses  of  these  three  elements  can- 
not be  taken  arbitrarily;  in  saying  that  these  simple  substances 
were  grouped  in  the  system  so  as  to  form  exclusively  calcium 
carbonate,  lime,  and  carbonic  acid  gas,  we  have  imposed  upon  these 
three  masses  a  certain  condition;  we  may  choose  arbitrarily  two  of 
these  masses,  the  mass  of  calcium  and  that  of  carbon,  for  example; 
but  the  third  mass,  that  of  oxygen,  is  then  determined  without 
ambiguity. 

We  may  always  group  the  simple  bodies  which  form  a  system 
of  a  given  kind  into  a  certain  number  of  components  or  of  residues 
of  components,  so  that  the  mass  of  each  of  these  groupings  may 
be  chosen  in  an  arbitrary  way,  without  contradicting  the  defini- 
tion itself  of  the  kind  of  systems  that  we  study;  thus,  in  saying 
that  the  systems  are  to  be  composed  of  calcium  carbonate,  lime, 
and  carbonic  anhydride,  we  have  defined  their  kind;  by  taking 
arbitrary  masses  of  calcium,  carbon,  and  oxygen,  we  could  not  in 
general  compose  from  them  a  system  of  the  kind  studied;  but 
in  taking  arbitrary  masses  of  lime  and  of  carbonic  anhydride,  we 
may  always  compose  such  a  system;  the  lime  and  carbonic  anhy- 
dride are,  for  the  kind  of  systems  studied,  independent  components. 

In  many  cases  it  is  possible  to  choose  in  many  different  ways 
the  independent  components  of  the  systems  of  a  given  kind;  for 
instance,  take  systems  formed  of  hydrated  crystals  of  sodium 
acetate  and  of  an  aqueous  solution  of  sodium  acetate;  we  may 
take  as  independent  components  of  the  systems  of  this  kind  water 
and  hydrated  sodium  acetate;  we  may  take  also,  for  independent 
components,  water  and  anhydrous  sodium  acetate.  But  if  the 
nature  of  the  independent  components  of  a  definite  kind  of  chemi- 
cal systems  may,  in  certain  cases,  show  a  certain  indefiniteness, 
the  number  of  these  independent  components  can  show  none;  it 
is  easy  to  demonstrate  the  following  theorem:  The  number  of 
independent  components  of  systems  of  a  given  kind  is  always  the 
same,  whatever  the  manner  of  grouping  the  independent  components 
of  the  elements  which  form  the  systems  of  the  kind  considered. 

Thus  in  the  svstems  that  we  have  taken  for  illustration  the 


108  THERMODYNAMICS  AND  CHEMISTRY. 

independent  components  may  be  chosen  in  two  different  ways; 
but  whichever  way  is  adopted,  their  number  remains  equal  to  2. 

It  is  in  general  a  very  easy  matter  to  determine  the  number 
of  independent  components  for  a  kind  of  chemical  system  when 
the  formulae  of  the  bodies  taking  part  are  known;  hereafter  we 
shall  denote  by  the  letter  c  the  number  of  independent  components 
which  form  the  systems  of  the  kind  studied. 

88.  Number  of  phases  into  which  a  given  system  is  divided. — 
The  heterogeneous  systems  studied  by  the  chemist  are  divided 
into  a  certain  number  of  homogeneous  masses;  a  system  formed 
of  water  and  water  vapor  is  divided  into  a  homogeneous  mass 
of  liquid  water  and  a  homogeneous  mass  of  water  vapor. 

Several  of  the  homogeneous  masses  which  compose  a  heteroge- 
neous system  may  have  the  same  nature,  the  same  chemical  and 
physical  properties;  in  a  bath  where  a  saturated  solution  of  sodium 
chloride  deposits  the  salt  that  it  contains,  the  crystals  of  salt 
adhere  to  various  parts  of  the  walls  of  the  bath;  although  sepa- 
rated from  each  other,  these  crystals  have  all  the  same  composition 
and  the  same  properties;  if  they  were  joined  together  they  would 
form  a  homogeneous  solid.  These  different  masses,  which,  although 
separated  from  each  other,  have  the  same  composition  and  proper- 
ties, belong  to  the  same  phase;  in  the  crystal  bath  of  which  we 
have  just  spoken,  the  salt  crystals  constitute  one  phase;  the  saline 
solution  constitutes  another  phase;  the  system  has  two  phases. 

It  is  in  general  very  easy  to  enumerate  the  distinct  phases 
which  are  encountered  in  a  given  system. 

A  homogeneous  mixture  contains  a  single  phase. 

A  system  formed  of  liquid  water  and  water  vapor  has  two 
phases,  the  liquid  and  the  vapor;  a  system  containing  salt  crystals 
and  a  solution  of  this  salt  has  likewise  two  phases,  the  solid  salt 
and  the  solution. 

A  system  formed  of  calcium  carbonate,  of  lime,  and  of  carbonic 
acid  gas  has  three  phases:  solid  calcium  carbonate,  solid  lime, 
carbonic  acid  gas. 

We  shall  denote  by  the  letter  <j>  the  number  of  phases  possessed 
by  any  system. 

89.  Fundamental  hypothesis. — Let  there  be  a  system  having 
the  <f>  phases  1,  2,  3, .  .  . ,  <£;  let  Mlt  M2, .  .  . ,  M^  be  the  masses 


THE  PHASE  RULE.  109 

of  these  various  phases;  the  analysis  of  J.  Willard  Gibbs  is  based 
on  the  following  principle: 

The  internal  thermodynamic  potential,  &,  of  the  system  is 
given  by  the  following  formula  : 

(1)  gr  =  M1*F1  +  MaSFa+  .  .  .  +M^. 

In  this  formula  3^  is  a  quantity  whose  value  depends  upon  the 
temperature,  upon  the  nature  of  the  phase  1,  of  its  composition,  of 
its  density;  we  may  say  that  (Ft  is  the  internal  thermodynamic 
potential  which  would  characterize  a  unit  mass  of  the  phase  1, 
if  we  considered  separately  this  mass  at  the  same  temperature,  in 
the  same  state,  with  the  same  composition  and  density  that  it 
has  in  the  midst  of  the  system.  Analogous  considerations  apply 

to  SF,  .  .  .  g> 

Is  the  truth  of  this  principle  evident  and  absolute?  Not  at 
all.  It  may  be  said  that  this  principle  regards  as  negligible  the 
forces  that  the  different  parts  of  the  system  exercise  on  each  other. 
This  hypothesis  may,  in  a  great  number  of  cases,  be  near  enough 
to  the  truth  so  that  the  consequences  deduced  from  it  accord  in 
a  sufficiently  satisfactory  manner  with  the  facts;  but  it  is  to  be 
expected  that  this  may  not  always  be  the  case;  we  shall  see,  in 
fact,  that  certain  phenomena  do  not  accord  with  the  laws  that 
we  are  going  >to  develop  ;  to  take  account  of  these  phenomena  it 
will  be  necessary  to  renounce  the  too  simple  principle  that  we 
have  just  stated  (see  Chap.  XVII). 

Nevertheless,  in  an  immense  number  of  cases,  this  principle  will 
lead  us  to  laws  which  will  be  in  most  satisfactory  accord  with  ex- 
periment; these  are  the  cases  of  which  we  shall  now  treat. 

90.  Variance  of  a  system.  —  From  the  point  of  view  of  sim- 
plified chemical  statics  to  which  we  shall  be  led  by  making  use  of 
the  hypothesis  implied  in  formula  (1),  a  system  is  characterized 
by  the  number  c  of  its  independent  components  which  form  it 
and  by  the  number  <j>  of  phases  into  which  it  is  divided;  in  a  still 
more  precise  manner,  the  form  of  the  law  of  equilibrium  for  a  chemi- 
cal system  depends  exclusively  upon  the  number 


(2)  F 

which  we  shaU  call  the  VARIANCE  of  the  system. 


110  THERMODYNAMICS  AND   CHEMISTRY. 

91.  Systems  of   negative  varian.e. — Let   us    enumerate   the 
forms  that  the  law  of  equilibrium  takes  according  to  the  value  of 
this  variance. 

In  the  first  place,  if  the  variance  of  the  system  is  negative,  the 
system  can  be  in  equilibrium  at  no  temperature  and  under  no  pressure; 
thus  it  would  be  impossible  to  have,  at  any  temperature  or  pressure, 
in  a  state  of  equilibrium  a  system  which  would  simultaneously 
include  sulphur  vapor,  liquid  sulphur,  the  orthorhombic  form  and 
the  monoclinic  form;  such  a  system  would  enclose  a  single  com- 
ponent divided  into  four  phases;  from  equation  (2)  the  variance 
would  be  —1. 

92.  Invariant   systems. — When  the  variance  of  a  system  is 
equal  to  zero  the  system  is  called  INVARIANT;   there  exists  but  one 
temperature  and  one  pressure  for  which  an  invariant  system  may  be 
in  equilibrium;  the  composition  and  the  density  of  each  of  the  phases 
which  compose  the  system  in  equilibrium  are,  besides,  determined; 
but  this  is  not  so  for  the  mass  of  each  phase;  even  if  the  total  mass 
of  each  of  the  independent  components  which  form  the  system  is 
given,  it  would  be  possible  to  divide  in  an  infinite  number  of  differ- 
ent ways   these  components  into  phases  having  the  composition 
proper  for  equilibrium. 

Consider,  for  example,  a  system  which  includes  water  simul- 
taneously in  each  of  its  three  states  of  ice,  liquid,  and  vapor;  this 
system  is  formed  of  a  single  independent  component  (c  =  l),  sepa- 
rated into  three  phases  (<£  =  3);  it  is  invariant;  it  can  therefore 
be  in  equilibrium  only  at  a  definite  temperature  and  at  a  definite 
pressure;  very  careful  investigations  have  shown  that  this  tem- 
perature, very  near  to  0°  C.,  has  the  value  +0°.0076  C.,  and  that 
this  pressure  is  the  tension  of  saturated  water  vapor  for  this  tem- 
perature, or  about  4.60  mm.  In  these  conditions  the  state  of  ice, 
liquid  water,  and  vapor  is  determined  without  ambiguity,  but 
this  is  not  so  for  the  mass  of  each  of  these  three  phases;  one 
may  without  disturbing  the  equilibrium  give  to  these  three  masses 
any  values,  provided  that  the  sum  of  these  three  values  remains 
constantly  equal  to  the  total  mass  of  the  system, 

93.  Monovariant  systems.    Transformation  tension  at  a  given 
temperature.     Transformation  point  under  a  given  pressure. — 
A  system  whose  variance  is  equal  to  unity  is  called  a  MONOVARIANT 


THE  PHASE  RULE.  Ill 

system.  In  order  to  have  a  mono  variant  system  in  equilibrium 
either  the  pressure  or  the  temperature  may  be  arbitrarily  chosen. 
At  a  given  temperature  the  pressure  for  which  the  system  is  in  equi- 
librium has  a  definitely  determined  value  which  is  called  the  TRANSFOR- 
MATION TENSION  at  the  temperature  considered.  The  composition 
and  the  density  of  each  of  the  phases  which  form  the  system  in 
equilibrium  are  likewise  determined;  as  with  the  transformation 
tension,  they  do  not  depend  upon  the  masses  of  the  independent 
components  which  constitute  the  system;  on  the  contrary,  the 
masses  of  the  various  phases  are  not  entirely  determined,  not 
even  when  the  masses  of  these  independent  components  are  given. 

Under  a  given  pressure  the  temperature  for  which  the  system  is 
in  equilibrium  has  a  definite  value,  which  is  called  the  transforma- 
tion point  under  the  pressure  considered;  this  equilibrium  tempera- 
ture does  not  depend  upon  the  masses  of  the  independent  com- 
ponents making  up  the  system,  and  it  is  the  same  for  the  compo- 
sition and  density  of  each  of  the  phases  of  the  system  into  which 
the  system  is  divided,  at  the  instant  of  equilibrium;  besides  this, 
the  masses  of  these  phases  are-  not  entirely  determined,  not  even 
when  the  masses  of  the  independent  components  are  given. 

94.  Examples  of  monovariant  systems. — The  most  simple 
type  of  monovariant  system  is  furnished  by  a  liquid  in  the  pres- 
ence of  its  vapor;  a  single  component  (c=l)  is  divided  into 
two  phases  (<£=2). 

For  a  given  temperature,  equilibrium  corresponds  to  a  com- 
pletely determined  value  of  the  pressure,  which  is  the  tension  of 
saturated  vapor  at  the  given  temperature;  the  densities  of  the 
liquid  and  of  the  vapor  have  definite  values  called  the  density 
of  saturated  liquid  and  the  density  of  saturated  vapor  at  this 
temperature;  on  the  other  hand,  the  mass  of  liquid  and  the 
mass  of  vapor  that  the  system  includes  are  not  determined;  we 
may  impose  upon  these  two  masses  all  the  variations  that  have 
their  sum  constant,  equal  to  the  total  mass  of  the  system. 

Under  a  given  pressure  there  is  an  equilibrium  temperature 
which  is  entirely  determined  by  the  knowledge  of  this  pressure; 
it  is  the  boiling-point  under  the  pressure  considered. 

A  system  which  includes  a  single  substance  at  once  in  the 
solid  and  liquid  states  is  also  a  monovariant  system;  under  a  given 


112  THERMODYNAMICS  AND  CHEMISTRY. 

pressure  equilibrium  corresponds  to  a  definite  temperature  which 
is  the  fusing-point  under  the  pressure  considered;  and  the  fusing- 
point  depends  upon  this  pressure  alone. 

A  system  which  contains  a  gas  such  as  cyanogen  and  a  poly- 
merous  solid  from  this  gas,  such  as  paracyanogen,  is  also  a  mono- 
variant  system;  similarly,  at  a  given  temperature,  it  is  necessary 
for  equilibrium  that  the  gas  reach  a  definite  tension;  this  tension, 
which  depends  upon  the  temperature  alone,  is  the  transforma- 
tion tension  for  this  temperature;  this  is,  in  fact,  the  law  found 
to  be  true  by  Troost  and  Hautefeuille  in  their  classic  investiga- 
tions. 

A  system  containing  calcium  carbonate,  lime,  and  carbonic 
acid  gas  consists  of  two  independent  components  (c=2)  divided 
into  three  phases  (<£  =  3);  it  is  a  monovariant  system;  at  a  given 
temperature  the  system  is  in  equilibrium  for  a  definite  value  of 
the  pressure,  called  the  dissociation  tension  of  calcium  carbonate 
at  the  given  temperature;  this  tension  depends  exclusively  upon 
the  temperature;  it  depends* in  no  wise  upon  the  masses  of  the 
independent  components,  lime,  and  carbonic  anhydride,  which 
make  up  the  system ;  this  is  the  celebrated  law  predicted  by  Henri 
Sainte-Claire  Deville,  demonstrated  by  Debray  for  the  case  that 
we  have  just  taken  as  example,  and  confirmed  by  Debray  and 
by  G.  Wiedemann  when  studying  the  dissociation  of  hydrated 
salts,  and  by  Isambert  from  a  study  of  the  dissociation  of  com- 
pounds that  ammonia  gas  forms  with  certain  metallic  chlorides. 

The  number  of  monovariant  systems  is  very  great.  An  anhy- 
dride or  a  hydrated  salt  is  taken  in  the  presence  of  a  water  solu- 
tion of  this  salt  in  the  presence  of  water  vapor;  two  independent 
components  (c=2),  the  salt  and  the  water,  exist  in  three  phases 
(<£  =  3),  the  solid  salt,  the  solution,  the  vapor;  the  system  is  mono- 
variant;  for  every  temperature  there  is  an  equilibrium  state  for 
the  system;  the  temperature  once  given,  the  tension  of  the  water 
vapor  and  the  concentration  of  the  solution  for  the  system  in  equi- 
librium have  definite  values. 

Chlorine  is  dissolved  in  water;  crystals  of  chlorine  hydrate 
are  deposited  from  the  solution,  which  has  above  it  a  gaseous 
atmosphere  which  is  a  mixture  of  chlorine  and  water  vapor;  two 
independent  components  (c=2),  water  and  chlorine,  exist  in  three 


THE  PHASE  RULE.  113 

phases  (<£  =  3),  the  solution,  the  chlorine  hydrate  crystals,  and  the 
gas  mixture;  the  system  is  therefore  monovariant;  for  to  each  tem- 
perature corresponds  a  state  of  equilibrium  of  the  system;  for  this 
state  of  equilibrium  the  tension  of  the  gas  mixture  is  determined, 
as  is  also  the  composition  of  the  liquid  mixture;  this  law  has  been 
verified  by  Isambert  and  by  H.  Le  Chatelier  for  mixtures  of  chlo- 
rine and  water;  it  has  been  verified  by  Wroblewski,  Bakhuis 
Roozboom,  and  P.  Villard  for  various  other  mixtures  with  gases 
which  form  hydrates. 

A  mixture  of  ether  and  water  separates  into  two  layers;  the 
one,  more  rich  in  ether  and  therefore  the  lighter,  floats,  while  the 
other,  richer  in  water,  occupies  the  bottom  of  the  vessel;  a  mixed 
vapor  is  above  the  liquids;  two  independent  components  (c=2), 
ether  and  water,  have  three  phases  (<£  =  3),  the  two  liquid  layers 
and  the  mixed  vapor;  the  variance  of  the  system  has  the  value  1; 
for  each  temperature  the  system  may  be  observed  in  equilibrium; 
and  it  is  sufficient  to  state  the  temperature  in  order  to  know  the 
tension  of  the  mixed  vapor  at  the  instant  of  equilibrium,  the  com- 
position of  the  vapor,  and  the  two  layers  of  liquid. 

These  examples,  that  might  be  multiplied,  suggest  the  infinite 
variety  of  types  of  monovariant  systems;  and  nevertheless,  in 
spite  of  the  diversity  of  these  types,  the  value  of  the  variance 
common  to  them  all  imposes  upon  them  all  the  same  form  of  the 
law  of  equilibrium;  in  all  we  find  a  transformation  tension  depend- 
ing solely  upon  the  temperature. 

95.  Role  of  monovariant  systems  in  the  history  of  chemical 
mechanics. — The  role  that  the  monovariant  systems  have  played 
in  the  history  of  chemical  mechanics  is  well  known;  it  is  because 
they  appealed  to  monovariant  systems  that  Debray,  Isambert, 
Troost,  and  Hautefeuille  found,  in  the  study  of  chemical  decom- 
positions, in  the  study  of  allotropic  modifications,  a  dissociation 
tension,  a  transformation  tension,  analogous  to  the  tension  of  satu- 
rated vapors;  it  is  in  showing  the  analogy  between  the  dissocia- 
tion tension,  the  transformation  tension,  and  the  tension  of  satu- 
rated vapor  that  they  have  made  even  the  most  skeptic  accept 
the  far-reaching  thought  of  Henri  Sainte-Claire  Deville:  There  is 
no  chemical  mechanics  distinct  from  physical  mechanics;  all 


114  THERMODYNAMICS  AND  CHEMISTRY. 

physical  changes  of  state  and  changes  of  chemical  "composition 
depend  upon  the  same  general  laws. 

96.  Bivariant  systems. — The  importance  of  monovariant  sys- 
tems should  not  make  us  forget  the  not  less  important  BIVARIANT 
systems. 

Such  are  called,  evidently,  those  systems  whose  variance  is 
2;  they  are  therefore  the  systems  existing  in  <j>  phases  equal  in 
number  to  the  independent  components  c  of  which  they  are  com- 
posed. 

A  bivariant  system  may  be  in  equilibrium  at  any  pressure  and 
at  any  temperature;  when  the  temperature  and  pressure  are  given, 
the  density  and  composition  of  each  phase  are  known;  they  depend 
in  no  wise  upon  the  masses  of  the  independent  components  of 
the  system;  furthermore,  if  these  masses  are  known,  the  mass  of 
each  of  the  phases  into  which  the  system  is  divided  is  in  general 
determined. 

A  very  simple  case  of  a  bivariant  system  is  furnished  by  a 
solid  salt  in  presence  of  an  aqueous  solution  of  this  salt;  two 
independent  components,  the  salt  and  the  water,  have  two  phases, 
the  solid  salt  and  the  solution.  For  every  temperature  and  pres- 
sure such  a  system  is  in  equilibrium;  the  solution  is  then  saturated 
with  the  salt;  the  concentration  of  the  saturated  solution  depends 
upon  the  temperature  to  which  it  is  brought  and  upon  the  pres- 
sure it  supports;  but  it  is  independent  of  the  masses  of  salt  and 
water  that  the  system  contains.  Also,  if  to  a  knowledge  of  the 
temperature  and  pressure  we  join  the  knowledge  of  the  total 
mass  of  the  salt  and  water  in  the  system,  the  masses  of  the  solu- 
tion and  of  the  undissolved  salt  are  determined. 

97.  Remark  on  the  law  of  equilibrium  of  bivariant  systems. 
—Here  we  must  guard  against  a  possible  confusion.  We  have 
said  that,  when  the  temperature  and  pressure  were  stated,  the 
concentration  of  the  saturated  solution  was  determined;  we  under- 
stand by  this  that  it  is  impossible,  at  a  given  temperature  and 
pressure,  to  find  a  series  of  saturated  solutions  such  that  the  con- 
centration varies  in  a  continuous  manner  from  one  solution  to 
the  following;  but  it  is  not  to  be  understood  that  the  constitution 
of  the  saturated  solution  is  determined  without  ambiguity;  it  may 
happen,  in  fact,  that  to  a  given  temperature  and  pressure  corre- 


THE  PHASE  RULE.  115 

spond  tun  distinct  concentrations  of  the  saturated  solution;  if, 
for  example,  we  studied  a  system  where  such  a  solid  hydrate  is 
in  presence  of  a  liquid  mixture  of  water  and  anhydride,  we  could, 
for  a  given  temperature  and  pressure,  obtain  two  saturated  solu-  , 
tions  of  distinct  compositions,  the  one  containing  more  water  than 
the  solid  hydrated  salt,  the  other  containing  less  water  than  this 
salt.  If,  besides  the  pressure  and  temperature,  the  total  mass 
of  the  anhyhride  and  that  of  the  water  which  compose  the  system 
are  given,  the  ambiguity  will  be  removed  and  the  equilibrium 
state  of  the  system  will  be  completely  determined. 

A  similar  remark  may  be  made  relative  to  the  composition  of 
each  of  the  phases  of  a  bivariant  system  in  equilibrium  at  a  given 
pressure  and  temperature;  it  is  a  remark  whose  importance  we 
shall  see  while  studying  in  Chap.  XI  the  indifferent  states  of  a  bi- 
variant system. 

The  water  solution  of  a  salt,  in  the  presence  of  this  solid  salt 
gave  us  a  first  example  of  a  bivariant  system.  For  another,  con- 
sider a  definite  mass  of  ether  into  which  it  poured  increasing  quan- 
tities of  water;  the  first  quantities  of  water  poured  in  mix  com- 
pletely with  the  ether;  but  beyond  a  certain  point  the  mixture 
divides  into  two  layers,  an  upper  one  richer  in  ether  and  a  lower 
richer  in  water;  we  therefore  have  to  do  with  a  system  formed 
of  two  independent  components,  ether  and  water,  and  divided 
into  two  phases,  the  two  superposed  liquid  layers;  such  a  system 
is  bivariant;  and  if  the  temperature  and  pressure  rest  constant, 
the  composition  of  the  two  liquid  layers  will  remain  invariable; 
as  water  is  little  by  little  added  to  the  mixture,  we  see  the  upper 
mass  decrease  and  the  lower  mass  increase,  but  neither  the  con- 
centration of  the  upper  nor  of  the  lower  layer  undergoes  any  change 
up  to  the  moment  when  enough  water  has  been  added  to  cause 
the  upper  layer  to  disappear;  the  system  will  then  cease  to  be 
bivariant. 

A  great  number  of  important  problems  in  chemical  statics  are 
dependent  upon  the  study  of  bivariant  systems.  The  theory  of 
the  solubility  of  gases  is  the  theory  of  a  bivariant  system;  for  the 
two  independent  components,  the  gas  and  the  solute,  exist  in  two 
phases,  a  liquid  solution  and  a  gaseous  atmosphere,  mixed  or 
simple  according  as  the  solute  is  volatile  or  not.  The  theory  of  the 


116  THERMODYNAMICS  AND  CHEMISTRY. 

vaporization  of  the  mixture  of  two  volatile  liquids,  the  theory  of 
the  liquefaction  of  the  mixture  of  two  gases  also  depends  upon  the 
study  of  a  bivariant  system,  for  two  fluids  which  play  the  role  of 
two  independent  components  are  divided  into  two  phases,  the 
liquid  and  vapor  mixtures. 

98.  There  are  contradictions  to  the  phase  rule. — What  we 
have  just  said,  and  also  what  we  shall  say  in  the  following 
chapter,  show  the  extreme  importance  of  the  phase  rule  in  chemi- 
cal mechanics. 

Does  it  follow  that  the  phase  rule  is  absolutely  true  and  that 
it  never  encounters  an  experimental  contradiction?  It  is  not  so. 
Observation  shows  us  a  considerable  number  of  facts  which  are 
irreconcilable  with  the  phase  rule  or  with  the  various  laws  that 
depend  upon  the  same  principles  as  this  rule. 

Let  us  consider  the  following  example,  which  later  we  shall 
consider  more  at  length: 

A  glass  tube  encloses  two  independent  components,  selenium 
and  hydrogen;  there  are  two  phases;  a  liquid  phase,  occupying 
the  lower  part  of  the  tube,  consists  of  selenium  which  has  dis- 
solved selenium  hydride;  a  gaseous  phase,  filling  the  upper  por- 
tion of  the  tube,  contains  hydrogen,  the  vapors  of  selenium  and 
of  selenium  hydride  gas. 

This  system,  consisting  of  two  components,  and  divided  into 
two  phases,  is  a  bivariant  system;  according  to  the  phase  rule  it 
may  be  in  equilibrium  at  every  pressure  and  temperature,  but 
once  the  temperature  and  pressure  fixed,  the  composition  of  each 
of  the  two  phases  of  the  system  is  determined.  This  is,  in  fact, 
the  law  obeyed  by  the  system  when  the  temperature  is  sufficiently 
high — when,  for  instance,  this  temperature  exceeds  350°  C. 

But  it  is  not  the  same  when  the  temperature  is  lower — when, 
for  instance,  it  is  equal  to  200°  or  250°  C.  When  the  pressure 
and  temperature  are  given,  the  composition  of  the  two  phases  in 
equilibrium  is  not  at  all  determined.  At  a  given  temperature  and 
pressure  the  gaseous  mixture  may  have  all  possible  compositions 
between  two  limiting  values,  the  one  rich  in  selenium  hydride, 
the  other  poor  in  this  substance.  Thus  there  may  be  an  infinity 
of  systems  in  equilibrium  which  should  not  be  the  case  accord- 
ding  to  the  phase  rule. 


THE  PHASE  RULE.  117 

99.  J.  Moutier's  rule  concerning  these  contradictions. — A 
great  number  of  analogous  examples  might  be  cited;  the  exami- 
nation of  all  these  cases  would  lead  to  the  following  conclusion, 
first  announced  by  J.  Moutier: * 

In  aU  cases  that  thermodynamics,  with  the  aid  of  the  principles 
and  hypotheses  mentioned  previously,  announces  that  a  certain  state 
Witt  be,  for  the  system  studied,  one  of  equilibrium,  experiment 
shows  that  the  system,  put  in  this  state,  will  remain  there  actually 
in  equilibrium.  But  when  thermodynamics  announces  that  the 
system  studied,  when  placed  in  a  certain  state,  will  undergo  a  definite 
modification,  it  may  happen  that  the  system,  put  in  this  state,  will 
remain  there  in  equilibrium. 

In  other  words,  experiment  always  recognizes  the  existence 
of  all  the  equilibrium  states  predicted  by  thermodynamics;  but  it 
recognizes,  besides,  the  existence  of  a  great  number  of  equilibrium 
states  which  contradict  the  predictions  of  thermodynamics.  For 
these  equilibrium  states  that  observation  recognizes,  while  ther- 
modynamics, such  as  we  have  developed  it  to  the  present  time, 
may  not  predict  the  existence,  we  shall  give  the  name  of  STATES 

OP  FALSE  EQUILIBRIUM. 

The  states  of  false  equilibrium  will  be  treated  at  length  towards 
the  end  of  this  work;  but  it  was  necessary  to  point  out  their  ex- 
istence now;  indeed,  we  shall  have  constantly  to  note  the  exist- 
ence of  these  states  when  we  wish  to  compare  the  indications  of 
theory  with  experimental  results. 

1  J.  MOUTIER,  Bulletin  de  la  Societe  philomathique,  7th  Series,  v.  4,  p.  86, 
1880 ;  Sur  quelyues  relations  de  la  Physique  et  de  la  Chimie  (Fr&ny's  Chemical 
Encyclopedia,  v.  2,  1881.) 


CHAPTER   VII. 
MULTIVARIANT  SYSTEMS. 

I.  TRIVARIANT  SYSTEMS. 

100.  Multivariant  systems.      Invariant  systems. — We  have 
just  described   briefly  the   monovariant    and   bivariant  systems. 
These  systems  are  at  once  the  simplest  and  the  most  important  in 
chemical  mechanics ;   hence  several  chapters  will  be  devoted  to 
them. 

For  the  moment  we  are  going  to  demonstrate  the  utility  of 
the  phase  rule  by  studying  the  multivariant  systems,  that  is  to  say, 
the  systems  whose  variance  is  at  least  equal  to  3.  The  com- 
plexity of  these  systems  is  in  general  so  great  that  they  could 
hardly  be  comprehended  without  the  aid  of  the  rule  for  which 
we  are  indebted  to  Gibbs. 

When  the  number  of  independent  components  exceeds  by 
unity  the  number  of  phases  into  which  the  system  is  divided 
(c=  </>  +  !),  the  variance  is  equal  to  3;  the  system  is  TRIVARIANT. 

In  order  to  know  completely  the  composition  of  the  phases 
into  which  a  trivariant  system  in  equilibrium  is  divided,  it  does 
not  suffice  to  know  the  temperature  and  the  pressure;  it  is  neces- 
sary to  add  a  third  quantity. 

101.  Theory  of  double  salts. — Let  us  consider  an  example  of 
a  trivariant  system. 

Two  anhydrides,  that  we  may  denote  by  1  and  2,  have  been 
dissolved  in  water,  denoted  by  0;  the  liquid  mixture  is  present 
with  a  solid  body  formed  by  the  combination,  in  definite  propor- 
tions, of  the  three  substances  0,  1,  2  or  of  some  of  them;  this  solid 
body  may  be  merely  ice  or  one  of  the  anhydrides  1  and  2;  it  may 
be  a  simple  hydra  ted  salt  formed  by  the  salt  1  or  by  the  salt  2; 
it  may  be,  finally,  a  double  salt,  anhydride  or  hydrate;  in  all  cases 

118 


MULTIVARIANT  SYSTEMS.  119 

the  system  is  formed  of  three  independent  components,  water  0 
and  the  salts  1  and  2,  divided  into  two  phases,  the  solid  and  the 
liquid. 

We  have  supposed  that  the  solid  state  was  a  definite  compound; 
the  composition  of  the  liquid  phase  may  therefore  alone  vary; 
we  may  represent  this  composition  by  means  of  the  two  concen- 
trations 

M  M 


Af  0,  Aft,  M2  being  the  three  masses  of  water  and  of  the  salts  1 
and  2  contained  in  the  solution. 

It  will  not  suffice,  in  order  to  know  the  values  of  the  two  con- 
centrations s1?  s2  of  a  solution  in  equilibrium  with  the  solid,  to 
know  the  temperature  and  the  pressure.  At  a  given  temperature 
and  pressure  an  infinity  of  values  of  the  concentrations  s1?  s2  may 
be  obtained,  for  which  there  is  equilibrium  between  the  solid 
body  and  the  solution;  if  it  is  desired  to  have  the  concentra- 
tions su  s2  determined,  it  is  necessary  to  join  to  the  temperature 
and  pressure  a  third  quantity  conveniently  chosen;  one  may, 
for  instance,  take  for  the  supplementary  known  quantity  one  of 
the  two  concentrations  «1;  s2;  the  other  is  then  entirely  deter- 
mined. 

102.  Surface  of  solubility  of  a  double  salt  at  a  given  pressure. 
—  Imagine  that  it  is  desired  to  study  a  similar  system  under  the 
constant  pressure  n,  equal,  for  example,  to  the  atmospheric  pres- 
sure; take  three  rectangular  axes  OT  ',  Ost,  Os2  (Fig.  22),  on  which 
we  lay  off  lengths  proportional  respectively  ss 
to  T,  slt  s2;  besides  the  pressure  TT,  let  us 
assume  a  value  of  the  temperature  T  and 


•— - M 


of  the  concentration  st;  these  values  deter- 
mine a  point  m  in  the  plane  TOs^  It 
follows,  when  TT,  T,  and  sl  are  given,  that 
the  value  assumed  by  the  concentration  s2  ° 
of  the  solution  in  equilibrium  with  the  solid 
body  is  determined;  through  the  point  m 
draw  a  line  mM  parallel  to  Os2  and  whose 
length  Is  proportional  to  s2;  the  point  M,  FIG,  22. 

which  has  the  coordinates  T,  sv  s2,  represents  the  composition  of 


120  THERMODYNAMICS  AND  CHEMISTRY. 

a  solution  susceptible  of  being  in  equilibrium  at  the  pressure  nt 
at  the  temperature  T,  with  the  solid  phase  considered.  For  every 
system  of  values  of  T  and  sl7  or,  in  other  words,  for  every  point 
ra  in  the  plane  T0s1  corresponds  similarly  a  point  M;  these 
points  taken  together  form  a  certain  surface  S;  each  point  in  this 
surface  represents  the  temperature  and  the  concentrations  of  a 
solution  that  may  be  observed,  under  the  pressure  n,  in  equilibrium 
in  contact  with  the  solid  body.  This  surface  S  separates  the  space 
into  two  regions;  every  point  situated  in  one  of  these  regions 
represents,  by  its  coordinates,  a  temperature  and  concentrations 
such  that,  under  the  pressure  TT  at  this  temperature,  a  solution 
having  these  two  concentrations  deposits  a  certain  quantity  of 
solid;  every  point  in  the  other  region  represents,  by  its  coordi- 
nates, a  temperature  and  concentrations  such  that,  under  the  pres- 
sure TT  at  this  temperature,  a  solution  having  these  two  concentra- 
tions is  capable  of  dissolving  a  new  quantity  of  the  solid  substance. 
The  surface  S  is  the  surface  of  solubility  of  the  solid  substance 
considered. 

It  is  also  easy  to  distinguish  the  •  region  of  space  whose 
various  points  represent  non-saturated  solutions  of  the  solid  C 
from  the  region  whose  various  points  represent  supersaturated 
solutions.  Thus  the  different  points  of  the  axis  OT  correspond 
to  s1  =  0,  s2  =  Q;  they  represent  therefore  pure  water,  which  evi- 
dently could  not  be  saturated  with  respect  to  the  substance  C 
except  in  the  particular  case  where  the  substance  C  is  ice;  the 
region  in  which  the  axis  OT  is  situated  is  therefore  the  region 
which  represents  non-saturated  solutions. 

103.  Case  in  which  the  solution  may  furnish  two  distinct 
salts.— It  generally  happens  that  a  solution  of  the  substances  1 
and  2  in  water  0  may,  according  to  circumstances,  deposit  different 
solid  bodies:  simple  salts  of  different  bases  or  acids,  salts  of  the 
same  base  and  of  the  same  acid,  but  differently  hydrated,  distinct 
double  salts,  etc.  Let  C  and  C'  be  two  different  solids.  For 
each  of  them  there  will  be  a  surface  of  solubility;  S  will  be  this 
surface  for  C,  and  S'  for  C'. 

Suppose  that  these  two  surfaces  S  and  S'  intersect  in  a  certain 
line  L  (Fig.  23).  The  line  L  will  separate  the  surface  S  into  two 
parts  Sl  and  S2,  and  the  surface  S'  into  two  parts  £/  and  £/. 


MULT1VARIANT  SYSTEMS. 


121 


Take  a  point  on  the  surface  S;  T,  sv  s2  are  its  coordinates- 
at  the  temperature  T,  a  solution  for  which  st  and  s2  are  the  con- 
centrations may  neither  dissolve  a  new  quantity  of  the  solid  C 
nor  let  this  solid  be  deposited.  Does  it  follow  that  a  system  con- 
sisting of  the  solid  C  and  the  solution  is  in  equilibrium?  This  is 


FIG.  23. 

not  assured,  for  it  may  happen  that  the  solution  deposits  a  cer- 
tain quantity  of  the  solid  C';  before  affirming  that  the  system  fo 
in  equilibrium,  we  must  be  sure  that  this  last  phenomenon  does 
not  take  place. 

The  surface  £'  separates  the  space  into  two  regions;  the  points 
in  the  first  region  represent  by  their  coordinates  T,  s1;  s2,  the  con- 
ditions in  which  the  solution  may  dissolve  the  solid  C'  but  not 
deposit  it;  the  points  in  the  second  region  represent  conditions 
in  which  the  solid  C'  cannot  stay  in  solution  but  may  be  precip- 
itated from  it. 

Of  the  two  parts  St  and  £>  into  which  the  line  L  divides  the 
surface  S,  one  of  them,  Sl}  is  in  the  first  of  these  regions  and  the 
other,  S2,  is  in  the  second. 

Take  a  point  of  coordinates  T,  slt  S2,  on  the  surface  S^  we 
know  that  at  a  temperature  T  a  solution  of  concentrations  slt 
S2  cannot — 

1°.  Dissolve  the  solid  C; 

2°.  Precipitate  the  solid  C; 

3°.  Precipitate  the  solid  C". 


122 


THERMODYNAMICS  AND  CHEMISTRY. 


The  point  considered  represents  therefore  conditions  where 
there  is  necessarily  equilibrium  in  a  system  which  contains  only 
the  solution  and  the  solid  C. 

Take,  on  the  contrary,  a  point  on  the  surface  *S2;  let  T,  slf 
s2  be  its  coordinates.  At  the  temperature  T  a  solution  of  con- 
centrations slf  s2  can  neither  dissolve  nor  precipitate  the  sub- 
stance C;  but  it  precipitates  the  substance  C";  a  system  which 
includes  only  the  substance  C  and  the  solution  is  not  in  equilib- 
rium in  the  conditions  represented  by  the  point  considered;  there 
is  formed  a  precipitate  of  substance  C'. 

We  should  find  similarly  that  of  the  two  parts  £/,  S2'  into 
which  the  line  L  divides  the  surface  Sr,  there  is  one,  as  £/,  for 
which  each  point  represents  a  state  of  equilibrium  for  a  system 
enclosing  only  the  solution  and  the  solid  C",  while  the  second,  S2', 
does  not  represent  equilibrium  states  for  such  a  system. 

To  sum  up,  if  it  is  desired  to  represent  the  conditions  (tem- 
perature and  concentrations)  in  which  may  be  observed  in  equi- 
librium the  solution  and  one  only  of  the  two  solid  deposits  C,  C', 
all  of  the  points  of  the  two  surfaces  of  solubility  S,  S'  should  not 
be  kept,  but  only  (Fig.  24)  the  points  of  one  part,  Sv  of  the  surface 


FIG.  24. 

S  and  of  a  part,  £/,  of  the  surface  S',  these  two  parts  having  for 
common  boundary  the  line  L;  if  the  point  chosen  is  on  the  surface 
S19  the  solid  deposit  will  be  formed  exclusively  of  the  substance 
C;  if  the  point  chosen  is  on -the  surface  £/,  the  solid  deposit  will 
consist  exclusively  of  the  substance  C'. 

104.  Conditions  in  which  the  two  precipitates  are  simultane- 
ously in  equilibrium  with  the  solution. — May  we  observe  a  sys- 


MULTIVARIANT  SYSTEAfS. 


123 


tern  in  equilibrium  enclosing  at  the  same  time  the  two  solid  pre- 
cipitates, C  and  C"  in  the  presence  of  the  solution?  For  this  it  is 
necessary  and  sufficient  that  the  solution  cannot — 

1°.  Dissolve  the  substance  C; 

2°.  Precipitate  the  substance  C; 

3°.  Dissolve  the  substance  C"; 

4°.  Precipitate  the  substance  C". 

The  first  two  conditions  require  that  the  representative  point 
be  on  the  surface  S,  and  the  last  two  that  this  point  be  on  the 
surface  $';  all  of  these  conditions  teach  us  that  the  represen- 
tative point  is  on  the  line  L,  the  intersection  of  the  surfaces  S 
and  S'.  The  line  L  is  therefore  the  locus  of  the  points  which 
represent  the  conditions  in  which  the  solution  may  be  observed 
in  contact  with  the  two  solid  deposits  C  and  C' 

When  the  solution  exists  in  the  presence  of  two  solid  deposits 
C,  C',  the  system,  formed  of  three  components,  is  divided  into 
three  phases;  it  is  therefore  Invariant;  let  us  show  that  the  results 
we  have  just  obtained  are  in  accord  with  the  properties  of  bi- 
variant  systems. 

Let  us  take  arbitrarily  a  pressure  TT  and  a  temperature  T. 
Construct  (Fig.  25)  the  two  surfaces  g 
S,  Sr,  which  correspond  to  the  pres- 
sure TT,  and  let  L  be  their  line  of  in- 
tersection. On  the  axis  of  tempera- 
tures take  a  point  T  whose  abscissa 
OT  corresponds  to  the  given  tem- 
perature, and  through  this  point 
draw  a  plane  7\7T2  parallel  to  the 
plane  sxOs2;  this  plane  cuts  the  line 
L  in  a  certain  point  P  whose  coor- 
dinates are  T,  sl}  s2;  under  the 
pressure  r.  at  the  temperature  T, 
the  solutions  of  concentrations  sl} 
S2  will  remain  in  equilibrium  in  contact  with  the  two  deposits 
C,  C".  We  see  therefore  that  at  each  pressure  and  temperature 
our  bivariant  system  may  have  an  equilibrium  state;  when  the 
pressure  TT  and  the  temperature  T  are  given,  the  composition  of 
each  phase  at  the  instant  of  equilibrium  is  determined. 


FIG.  25. 


124  THERMODYNAMICS  AND   CHEMISTRY. 

105.  Case  in  which  the  solution  may  give  three  distinct 
salts. — It  may  be  that  the  solution  considered  may  precipitate 
not  only  two  but  three  solids,  C,  C',  C" .  We  shall  therefore  have 
to  distinguish  three  kinds  of  equilibrium  states  of  the  system: 

1°.  Equilibrium  of  the  system  formed  by  the  solution  and  a 
single  solid  precipitate. 

At  the  pressure  considered,  the  conditions  in  which  such  a 
state  of  equilibrium  may  be  observed  are  represented  by  the  three 

co-ordinates  of  a  point  situated  on  one 
of  the  three  surfaces  S,  S'}  S"  (Fig.  26), 
which  are  parts  of  the  solubility  surfaces 
of  the  substances  C,  C',  C",  respectively; 
according  as  the  representative  point 
is  situated  on  the  surface  S,  S',  or  S", 
J_  the  solid  deposit  consists  exclusively  of 
the  substances  C,  C',  or  C". 

2°.  Equilibrium  of  the  system  formed 
by  the  solution  of  two  solid  deposits. 

The  representative  point  will  be  on 
one  of  the  three  lines  L,  Z/,  L" ',  bounda- 
ries of  the  surfaces  S,  S',  S";  if  the  two 

solid  deposits  are  the  substances  C',  C",  the  representative  point 
will  be  on  the  line  L,  boundary  of  the  two  surfaces  S',  S";  if  the 
two  solid  deposits  are  the  substances  C",  C,  the  representative 
point  will  be  on  the  line  I/,  boundary  of  the  surfaces  S",  S;  if 
the  two  solid  deposits  are  the  substances  C.  Cf,  the  representa- 
tive point  will  be  on  the  line  L",  boundary  of  the  two  surfaces 
S,  S'. 

3°.  Equilibrium  of  the  system  formed  by  the  solution  and 
the  three  solid  deposits. 

The  representative  point  will  be  the  point  P  common  to  the 
three  surfaces  S,  S',  S",  and  hence  of  the  three  lines  L,  I/,  L" . 
Under  the  pressure  considered,  there  is  thus  a  single  temperature 
and  a  single  composition  of  the  solution  for  which  such  an  equi- 
librium is  possible,  a  condition  that  should  not  astonish  us;  the 
system,  formed  of  three  independent  components,  is  then  divided 
into  four  phases;  it  is  monovariant. 


MULTIVARIANT  SYSTEMS.  125 

106.  The  Alloy:    Lead,  tin,  bismuth.     Charpy's  Researches. 

— A  chemical  system  which  gives  a  very  clear  illustration  of  the 
preceding  considerations  has  been  studied  recently  by  G.  Charpy.1 
The  liquid  phase  consists  of  a  mixture  of  the  three  metals,  lead, 
tin,  and  bismuth,  in  the  state  of  fusion,  a  mixture  that  may  be 
compared  to  the  solution  of  which  we  have  just  spoken  when  we 
attribute  the  three  indices  0,  1,  2  to  the  three  metals  studied, 
taken  in  any  order. 

The  representative  points  of  the  states  of  equilibrium  between 
the  mixture  in  fusion  and  the  solid  lead  form  a  surface  S;  the 
representative  points  of  equilibrium  states  between  the  fused 
mixture  and  the  solid  tin  form  a  surface  Sr;  finally,  the  represen- 
tative points  of  the  equilibrium  states  between  the  fused  mixture 
and  the  solid  bismuth  form  a  surface  S";  these  three  surfaces  S, 
S',  S"  have  been  constructed  by  G.  Charpy. 

The  points  on  the  line  L  represent  the  conditions  in  which  the 
liquid  mixture  may  be  in  equilibrium  with  solid  bismuth  and  solid 
tin;  the  points  on  the  line  L'  represent  the  conditions  in  which 
the  liquid  mixture  may  be  in  equilibrium  with  solid  lead  and 
solid  bismuth;  the  points  on  the  line  L"  represent  the  conditions 
in  which  the  liquid  mixture  may  be  in  equilibrium  with  solid 
lead  and  solid  tin. 

Finally,  the  coordinates  of  the  point  P  represent  the  values  that 
must  be  given  to  the  temperature  and  composition  of  the  liquid 
mixture  in  order  that  the  latter  may  remain  in  equilibrium  in  con- 
tact with  the  three  metals  in  the  solid  state.  According  to  Charpy's 
researches  the  value  of  this  temperature  is  +96°  C.  and  the  liquid 
mixture  corresponding  to  the  point  P  has  the  following  compo- 
sition : 

Lead 0.32 

Tin 0.16 

Bismuth 0 . 52 

107.  Mixture  of  three  melted  salts. — Hector  R.  Caweth 2  has 
studied  an  analogous  system;  but  here  the  liquid  mixture,  instead 

1  G.  CHARPY,  Comptes  Rendus,  v.  126,  p.  1569, 1898. 

2  H.  R.  CAWETH,  Journal  of  Physical  Chemistry,  v.  2,  p.  209,  1889. 


126  THERMODYNAMICS  AND  CHEMISTRY. 

of  being  formed  of  three  melted  metals,  consisted  of  three  nitrates 
in  fusion,  the  nitrates  of  potassium,  sodium,  and  lithium. 

108.  Domain  of  a  precipitate. — In  a  great  number  of  cases 
the  distinct  solid  precipitates  that  are  encountered  are  much 
greater  than  three  in  number.  Nevertheless  the  properties  of  the 
system  may  be  studied  and  represented  in  accordance  with  the 
principles  just  developed. 

Let  C,  C',  C"  .  .  .  be  the  precipitates  that  may  be  observed. 
Under  the  given  pressure  TT  the  states  of  equilibrium  between  the 
liquid  mixture  of  the  three  independent  components  and  the  single 
solid  precipitate  C,  states  in  which  the  system  is  trivariant,  are 
represented  by  the  various  points  on  a  limited  surface  S  that  is 
called  the  domain  of  the  precipitate  C. 

This  surface  S  touches  other  surfaces,  S',  S"  .  .  . ,  which  are  the 
domains  of  the  precipitates  C',  C"  .  .  .  Thus  the  possible  equi- 
librium states,  under  the  pressure  considered,  between  the  liquid 
mixture  and  a  single  solid  precipitate  are  represented  by  the  various 
points  of  a  polyhedral  surface  with  curved  faces  which  has  as  many 
pices  as  there  are  distinct  solid  precipitates  able  to  exist  within 
the  liquid  mixture. 

The  points  on  the  intersections  of  these  faces  represent  a  state 
of  equilibrium,  under  the  given  pressure,  between  the  mixture 
and  two  distinct  solid  precipitates;  for  these  states  the  system  has 
become  bi variant;  the  bounding  line  between  the  domain  of  the 
precipitate  C  and  that  of  C'  represents  all  the  possible  states  of 
equilibrium  between  the  liquid  mixture,  the  precipitate  C,  and 
the  precipitate  C'. 

Each  of  the  cusps  of  this  surface  represents  a  state  of  equilib- 
rium for  which  the  system,  become  monovariant,  is  composed  of  a 
liquid  mixture  and  of  three  solid  precipitates  C,  C',  C",  which  are 
those  whose  domains  S,  *S',  S"  join  at  the  cusp  considered. 

Finally,  it  is  not  necessary  to  consider  states  of  equilibrium 
for  which  the  liquid  mixture  exists  in  the  presence  of  three  solid 
precipitates;  the  system  would  then  be  invariant;  it  could  be 
in  equilibrium  only  at  a  single  temperature  and  pressure;  the 
case  in  which  this  pressure  would  be  the  one  given  is  evidently 
exceptional. 

It  is  easv  to  see  what  services  such  a  system  of  representation 


MULTIVARIANT  SYSTEMS. 


127 


may  render;  it  makes  easy  the  prediction  which  precipitates  may, 
in  the  given  conditions,  exist  in  presence  of  a  liquid  mixture  of 
three  independent  components. 

109.  System  composed  of  water,  ferric  chloride,  hydro- 
chloric acid.  Researches  of  Bakhuis  Roozboom  and  Schreine- 
makers. — H.  W.  Bakhuis  Roozboom  and  Schreinemakers l  have 
used  this  method  to  represent  the  various  states  of  saturation  of 
a  liquid  mixture  formed  of  the  three  following  components: 

Water:  H2O 

Hydrochloric  acid :  HC1 
Ferric  chloride:         Fe^Cl^ 

The  solid  precipitates  which  were  observed  in  these  investiga- 
tions are  twelve  in  number,  as  follows: 

Ice:  H20 

3  hydrates  of  hydrochloric  acid:  HC1  •  3H2O 

HC1-2H.O 
HC1-  H2O 
Anhydric  ferric  chloride:  Fe2Cl6 

4  hydrates  of  ferric  chloride:         Fe2Cl0-  12H2O 

Fe2CV  7H20 
Fe2Cle-  5H2O 
Fe2Cl«-  4H2O 

3  tertiary  compounds :  Fe2Clc  •  2HC1  •  12H2O 

Fe2Cle-    2HC1-  8H2O 
Fe2Cl6-   2HC1-  4H2O 

It  is  evident  how  difficult  it  would  have  been  to  unravel  the 
possible  equilibrium  states  from  such  a  system  without  the  aid 
of  the  theoretical  principles  previously  developed. 

no.  System:  water,  potassium  sulphate,  and  magnesium 
sulphate.  Investigations  of  Van  der  Heide.— These  principles 
have  been  of  service  in  the  study  of  not  less  complicated  systems. 

Van  der  Heide 2  has  applied  them  to  the  study  of  systems 
whose  three  independent  components  are: 

1  H.  W.  BAKHUIS  ROOZBOOM  and  SCHREINEMAKERS,  Zeit.  fur  physi- 
kalische  Chemie,  v.  15,  p.  588,  1894. 

3  VAN  DER  HEIDE,  Zeit.  fur  physikalische  Chemie,  v.  12,  p.  416,  1893. 


128  THERMODYNAMICS  AND  CHEMISTRY. 

Water:  H2O 

Potassium  sulphate :    K2S04 
Magnesium  sulphate :  MgSO4 

At  a  temperature  less  than  100°  C.  the  six  following  precip- 
itates may  be  obtained: 

Ice:  H20 

Anhydrous  potassium  sulphate:  K2SO4 

2  hydra  ted  sulphates  of  magnesium :    MgSO4  •  7H20 

MgSO4-6H2O 
2  double  salts :  MgK2(SO4) .  6H2O  (Schcenite) 

MgK2(SO4)-4H2O  (Leonite) 

The,  various  equilibrium  states  that  may  be  had  at  temperatures 
less  than  100°  are  represented  by  the  surfaces  whose  general  ap- 


PIG.  27. 

pearance  is  given  in  Fig.  27.     The  hidden  face  GGff^g  is  the 
domain  of  ice. 

in.  System:    water,   potassium   chloride,    and   magnesium 
chloride.      Researches     of    Van't    Hoff     and    Meyerhoffer.^- 

Van't  Hoff  and  Meyerhoffer l  have    studied  in    the   same   way 

1  VAN'T  HOFF  and  MEYERHOFFER,  Sitzungsberichte  der  Berliner  Akademie, 
1897,  p.  487;  Zeit.  fur  phys.  Chemie,  v.  30,  p.  64,  1899. 


MULTIVARIANT  SYSTEMS. 


129 


the   system   formed  of   the   following   three   independent   com- 
ponents: 

Water:  H2O 

Potassium  chloride:    KC1 

Magnesium  chloride :  MgCl2 

The  studies  of  Van't  Hoff  and  Meyerhoffer  extended  to  about 
185°.  In  these  conditions  one  may  obtain  the  following  eight 
distinct  precipitates: 

Ice:  H20 

Anhydrous  potassium  chloride :          KC1 

5  hydrates  of  magnesium  chloride:  MgCl2.12H2O 

MgCl2-  8H20 
MgCl2-  6H2O 
MgCl2-  4H20 
MgCl2-  2H20 

A  double  salt:  MgKCl3-6H2O    (Carnallite) 

Fig.  28  gives  a  general  idea  of  the  aspect  of  the  surface  which 


MgCl2,2HjO 


FIG.  2& 
represents  the  states  of  equilibrium  of  such  a  system. 


130  THERMODYNAMICS  AND  CHEMISTRY. 

II.    QUADRI  VARIANT    SYSTEMS. 

112.  Quadrivariant  systems  formed  of  four  components 
divided  into  two  phases.  —  If  four  components  exist  in  two  phases, 
we  have  c=4,  $  =  2, 


and  the  system  is  QUADRIVARIANT.  When  the  values  of  the 
pressure  it  and  of  the  temperature  T  are  given,  the  quadrivariant 
system  may  be  observed  in  equilibrium;  but  the  composition  of 
the  two  phases  in  equilibrium  is  far  from  being  determined  by 
the  knowledge  of  the  values  of  x  and  T  alone;  to  these  quantities 
it  is  necessary  to  add  two  others  in  order  to  fix  the  composition 
of  the  system  in  equilibrium. 

Let  us  take  an  example.  A  system  is  composed  of  four  inde- 
pendent components:  A  solute  0,  water,  for  instance,  and  three 
salts,  1,  2,  and  3.  This  system  is  separated  into  two  phases,  one 
liquid  and  one  solid  phase;  the  liquid  phase  is  composed  of  a  mass 
M0  of  water  and  of  masses  Mlt  M2,  M3  of  salts  1,  2,  3;  the  three 
concentrations  of  this  solution  are 


_ 

I~MO'    2~M0'    3  MO' 

The  solid  phase  is  a  body  of  definite  composition  formed  at  the 
expense  of  the  components  0,  1,  2,  3:  ice,  simple  salt  —  anhydrous 
or  hydrous  —  double  anhydrous  or  hydrous  salt;  we  shall  denote 
this  solid  by  the  letter  C. 

When  the  pressure  n  and  the  temperature  T  are  given,  the 
constitution  of  the  solution  capable  of  remaining  in  equilibrium 
in  contact  with  the  substance  C  is  not  yet  determined;  an  infinity 
of  different  solutions  may,  under  this  pressure  and  at  this  tem- 
perature, remain  in  equilibrium  in  contact  with  the  substance  C, 
without  dissolving  it  and  without  letting  a  new  mass  of  this  body 
be  precipitated.  To  the  knowledge  of  the  pressure  it  and  the 
temperature  T  let  us  join  the  values  of  two  of  the  concentrations 
«i>  S2>  ss>  then,  and  then  only,  shall  we  know  the  value  which  the 
third  concentration  should  have  in  order  that  there  may  be  equi- 
librinm  between  the  solution  and  the  substance  C. 


MULTIVARIANT  SYSTEMS.  131 

113.  Three  salts  dissolved  in  water.  Solubility  surface  of  a 
precipitate  at  a  given  pressure  and  temperature. — Let  us  try  to 

represent  all  the  compositions  for  which  the  solution  may  be  in 
equilibrium  with  the  substance  C  under  the  pressure  n  whose  value 
is  chosen  once  for  all,  atmospheric  pressure  for  example,  and  at 
a  temperature  whose  value  is  also  fixed,  as  +15°  C. 

On  three  axes  of  rectangular  coordinates  Os1;  Os,,  Os3  (Fig.  29) 
lay  off    lengths  proportional  Ho   the  three 
concentrations  slf  s2,  s3  respectively. 

Suppose  given  two  of  these  concentra- 
tions, as  sl  and  s2;  and  let  m  be  the  point 
in  the  plane  «10s2,  whose  coordinates  Oslr 
Os2  represent  these  two  concentrations.  At 
these  two  concentrations  it  is  necessary  and 
sufficient,  in  order  to  assure  equilibrium 
between  the  solution  and  the  substance  C, 
under  atmospheric  pressure  and  at  the' 
temperature  15°  C.,  to  add  a  well-deter-  FIG.  29. 

mined  value  of  the  concentration  ss;  through  the  point  m  draw 
a  parallel  mM  to  Os3  whose  length  mM=Os3  is  measured  by  this 
value  Os,;  the  three  coordinates  of  the  points  M  represent  the 
three  concentrations  of  a  solution  capable  of  remaining  in  equi- 
librium in  the  presence  of  the  substance  C  at  atmospheric  pressure 
and  a  temperature  of  15°. 

sx  and  s2  may  be  taken  arbitrarily,  or,  in  other  words,  the  point 
m  may  be  taken  anywhere  in  the  plane  s^s*;  for  every  position 
of  the  point  m  corresponds  a  point  M  on  the  surface  S. 

Each  point  M  of  the  surface  S  represents,  by  its  three  coordi- 
nates, the  three  concentrations  of  a  solution  susceptible  of  re- 
maining in  equilibrium,  under  atmospheric  pressure  and  at  15°, 
in  contact  with  the  substance  C. 

This  surface  S  divides  the  space  into  two  regions.  Every 
point  of  one  of  these  two  regions  represents,  by  its  three  coordi- 
nates, the  three  concentrations  of  a  solution  capable  of  dissolving 
a  certain  quantity  of  the  substance  C  under  atmospheric  pressure 
and  at  15°  temperature. 

Each  point  in  the  other  region  represents  by  its  coordinates 


132  THERMODYNAMICS  AND   CHEMISTRY. 

the  three  concentrations  of  a  solution  capable  of  precipitating  a 
certain  quantity  of  the  substance  C  under  the  same  conditions. 

•  The  origin  of  the  coordinates  for  which  Sj  =  0;  s2  =  0,  <s3=0, 
represents  pure  water.  If  the  solid  C  is  not  ice,  pure  water  could 
not  be  saturated  with  the  solid  C;  the  origin  of  the  coordinates 
is  therefore,  with  respect  to  the  surface  S,  within  the  region 
which  represents  the  unsaturated  solutions  of  the  solid  C;  this 
property  allows  of  recognizing  this  region  at  once. 

In  general  a  solution  formed  of  four  independent  components 
may  precipitate  various  solids  of  definite  composition,  as  C,  C', 
C".  .  .  .  By  a  method  similar  to  that  used  for  tri variant  systems, 
we  may  reach  the  following  conclusions: 

The  three  concentrations  of  a  solution  capable  of  remaining 
in  equilibrium,  under  atmospheric  pressure  and  at  15°  C.,  in  con- 
tact with  a  single  one  of  the  solids  C,  C',  C"  .  .  . ,  are  represented 
by  the  three  coordinates  of  a  point  M  belonging  to  a  certain  sur- 
face. 

This  surface  is  formed  of  a  certain  number  of  curved  areas 
S,  Sf,  S"  .  .  .  bounded  by  separating  lines;  in  other  terms,  it 
forms  a  polyhedral  surface  with  curved  faces. 

Each  of  the  faces  S,  S',  S"  .  .  .  of  this  polyhedral  surface 
corresponds  to  one  of  the  bodies  C,  C',  C"  .  .  .  and  forms  its 
domain. 

If  the  representative  point  M  belongs  to  the  domain  S  of  the 
body  C,  the  solution  which  has  for  concentrations  the  three  coor- 
dinates of  the  point  M  rests  in  equilibrium,  under  atmospheric 
pressure  and  at  15°  C.,  in  contact  with  the  substance  C,  but  not 
in  contact  with  another  precipitate. 

If  the  representative  point  is  on  the  boundary  line  of  the  two 
surfaces  S,  S'  of  the  two  substances  C,  C",  the  three  coordinates 
of  this  point  represent  the  three  concentrations  of  a  solution  which 
may,  under  atmospheric  pressure  and  at  15°  C.,  remain  in  equi- 
librium in  contact  with  a  solid  precipitate  composed  of  C  and  C"; 
in  such  a  state  of  equilibrium  one  system  of  four  independent 
components  is  divided  into  three  phases,  so  that  it  is  no  longer 
quadri variant  but  trivariant. 

Finally,  if  the  representative  point  is  at  the  cusp  where  the 
domains  S,  S',  £"  of  the  three  substances  C,  C",  C"  intersect,  the 


MULTIVARIANT  SYSTEMS.  133 

three  coordinates  of  this  point  represent  the  three  concentrations 
of  a  solution  capable  of  remaining  in  equilibrium,  under  atmos- 
pheric pressure  and  at  15°  C.,  in  contact  with  a  precipitate  which 
contains  the  three  substances  C,  C',  C"  mixed  together.  In  such 
a  state  the  system  formed  of  four  independent  components  and 
divided  into  four  phases  has  become  bivariant;  also,  the  pres- 
sure and  temperature  being  given,  the  composition  of  each  phase 
is  determined. 

114.  System:  Water,  magnesium  chloride,  magnesium  sul- 
phate, chloride  of  potassium,  potassium  sulphate.  Investiga- 
tions of  Loewenherz,  Van't  Hoff,  and  Meyerhoffer.—  An  inter- 
esting example  of  these  various  considerations  is  found  in  the 
researches  of  Loewenherz,1  continued  recently  by  Van't  Hoff, 
Meyerhoffer,  and  Don  nan.2 

With  the  object  of  analyzing  the  conditions  in  which  some  of 
the  numerous  compounds  are  formed  in  the  deposits  of  Stassfurt 
salt,  these  authors  have  studied,  under  atmospheric  pressure  and 
at  15°  C.,  the  solutions  formed  by  mixing  with  water,  H2O,  potas- 
sium chloride,  KC1,  magnesium  sulphate,  MgSO4,  and  magnesium 
chloride,  MgCl2. 

If  desired,  the  system  may  be  regarded  as  made  up  of  the 
four  independent  components 

(1)  H20,    KC1,    MgS04,    MgCL. 

But  we  shall  find  it  advantageous  to  make  another  choice. 
Within  the  solutions  there  may  be  formed  by  double  decom- 
position, potassium  sulphate,  KjSO4,  as  given  by  the  equation 

(2)  2KC1  -f  MgS04 = MgCl2  +  K2SO4. 

There  is  therefore  no  reason  for  considering  that  the  solution 
contains  actually  the  substances  indicated  in  equation  (1). 

Not  to  make  any  hypothesis,  we  shall  consider  the  solution 
as  being  the  mixture,  in  any  state  whatever,  of  the  four  following 
bodies : 

1  LCEWENHERZ,  Zeit.  fur  physikalische  Chemie,  v.  12,  p.  459,  1894. 

'VAN'T  HOFF  and  MEYERHOFFER,  Sitzungsber.  d.  Berliner  Akad.,  1897, 
p.  1019;  VAN'T  HOFF  and  DONNAN,  ibid.,  p.  1146;  VAN'T  HOFF,  Report  to 
International  Physics  Congress,  Paris,  1900,  v.  i,  p.  464. 


134  THERMODYNAMICS  AND  CHEMISTRY. 

(3)  H20,     C12,     K2,     S04,     Mg. 

We  denote  by  TT  the  molecular  weight  of  water  (18  grammes) 
and  by  w,  w2,  w3,  wr  the  number  of  grammes  that  represent  the 
symbols  of  the  four  other  bodies. 

In  a  molecule  (or  18  grammes)  of  water  of  a  given  solution 
chemical  analysis  gives  us 

nw    grammes  of  chlorine, 
n2w2  grammes  of  potassium, 
n.,Wj  grammes  of  S04, 
n'wf  grammes  of  magnesium. 

The  composition  of  the  solution  is  therefore  known  if  we  know 
the  four  numbers  n,  n2,  n3,  n'. 

But  it  is  not  necessary  to  know  these  four  numbers;  if  three 
of  them,  n,  n3,  n'  ,  are  known,  the  knowledge  of  the  fourth,  n2t 
follows. 

Thus  the  n2w2  grammes  of  potassium  consist  of  p2w2  grammes 
united  with  p2w  grammes  of  chlorine,  and  of  q2w2  grammes  united 
with  q2w3  grammes  of  SO4: 


Also,  the  n'w'  grammes  of  magnesium  consist  of  p'w'  grammes 
joined  to  p'w  grammes  of  chlorine,  and  of  q'wf  grammes  joined 
to  q'w3  grammes  of  S04: 

n'=p'+q'. 

Expressing  the  fact  that  the  chlorine  united  with  potassium 
and  with  magnesium  gives  for  the  total  quantity  of  chlorine: 

p2w+p'w=nw,    or    n= 


Writing  that    S04  united  with  potassium   and  S04  united  with 
magnesium  form  the  totality  of  SO4  : 


=  n3w3,    or    n3  = 
These  four  equations  give  us 
(4)  n+n3=rc2+n', 

so  that  when  n,  n3,  and  nf  are  known  it  is  easy  to   calculate  n2. 
One  may  choose  arbitrarily,  therefore,  the  masses  of  four  of  the 


MULTIVARIANT  SYSTEMS.  135 

substances  in  group  (1);  the  mass  of  the  fifth  is  determined; 
of  the  five  bodies  of  group  (1),  four  only  are  independent;  by 
changing  the  choice  of  independent  components  we  have  not 
altered  their  number  (see  Art.  87). 

But  in  order  that  the  system  may  be  taken  as  formed  by  the 
independent  components  (1),  the  numbers  n,  n2,  n3,  n'must  satisfy 
a  certain  condition. 

All  the  potassium  contained  in  18  grammes  of  water  will  have 
been  brought  there  by  the  chloride  of  potassium  dissolved  in  it. 
If  we  denote  by  n2  the  number  of  grammes  represented  by  2KC1 
to  obtain  n2w2  grammes  of  potassium,  it  would  be  necessary  to 
use  njr2  grammes  of  potassium  chloride  bringing  n2w  grammes 
of  chlorine. 

This  chlorine  is  not  all  the  chlorine  that  18  grammes  of  water 
contain;  we  must  take  into  account  that  from  the  magnesium 
chloride;  if  ^  is  the  number  of  grammes  represented  by  MgCl2, 
and  if  in  18  grammes  of  water  we  have  dissolved  n^  grammes 
of  this  substance,  which  would  bring  njjo  grammes  of  chlorine,  we 
have 

n2w+nlw=nw, 
or 

(5)  n^n-n^', 

and  as  nt  cannot  be  negative,  we  see  that  for  a  system  to  be  formed 
of  the  substances  (1),  it  is  necessary  that  the  numbers  n,  n^,  1%,  n'. 
satisfy  not  only  equation  (4),  but  also  the  condition 

(6)  n-n,^0. 

This  is  sufficient  also.  Let  us  denote  by  x3  the  number  of 
grammes  of  MgSO4;  in  dissolving  in  18  grammes  of  water  n#:2 
grammes  of  potassium  chloride,  n37r3  grammes  of  magnesium  sul- 
phate, and  n1^1  =  (n— n2)^  grammes  of  magnesium  chloride,  we 
shall  obtain  the  composition  sought 

Let  us  consider  now  the  systems  formed  by  the  four  following 
components : 

(!')  H20,    KC1,    MgS04,    K2S04. 


136  THERMODYNAMICS  AND  CHEMISTRY. 

They  may  also  be  considered  as  formed  of  the  five  components 
(3);  here  again  the  five  components  will  not  be  independent,  for 
the  condition  (4)  will  still  hold.  This  will  not  suffice  in  order  that 
the  system  may  be  considered  as  formed  by  the  components  (I'); 
a  condition  must  be  joined  to  this. 

In  18  grammes  of  water  we  put  nw  grammes  of  chlorine;  all 
of  this  substance  comes  from  potassium  chloride;  there  must 
have  been  dissolved,  therefore,  n7T2  grammes  of  potassium  chloride; 
but  this  is  not  all  the  potassium  contained  in  18  grammes  of  water; 
if  there  are  TT/  grammes  of  K2SO4,  and  if  in  18  grammes  of  water 
there  are  dissolved  W/TT/  grammes  of  potassium  sulphate,  there 
have  been  introduced  n^fw2  grammes  of  potassium.  We  have, 
therefore, 

nw2+n'w2=n2w2, 
or 

(5')  <=n2-n. 

As  n/  cannot  be  negative,  we  must  have 

(60  n-n2<0. 

This  suffices  also;  for  if  this  condition  holds,  it  is  only  neces- 
sary, as  is  evident,  to  dissolve,  in  18  grammes  of  water,  n.27T2 
grammes  of  potassium  chloride,  n3ns  grammes  of  magnesium  sul- 
phate, and  n1/7r/=  —  (n  —  n2)7r/  grammes  of  potassium  sulphate,  in 
order  to  obtain  a  solution  of  the  composition  indicated. 

.We  see,  therefore,  that  in  studying  the  systems  formed  by  the 
five  components  (3),  only  four  of  which  are  independent,  in  virtue 
of  equation  (4)  we  are  studying  at  the  same  time  the  systems  (1) 
and  the  systems  (10;  we  have  to  do  with  the  first  if  (n—n2)  is 
positive,  and  with  the  second  if  (n-n2)  is  negative. 

In  order  to  represent  the  composition  of  a  solution,  it  suffices 
by  equation  (4)  to  know  the  value  of  three  of  the  numbers  n,  n^, 
n^  nr,  or,  if  it  is  preferred,  the  three  values 

(7)  x=n-n2,    y=n2}    z=n3. 

Hence  the  constitution  of  a  solution  may  be  represented  in 
the  following  manner: 


MULTIVARIANT  SYSTEMS. 


137 


X 


Let  us  take  (Fig.  30)  a  system  of  rectangular  coordinates  Ox, 
Oy,  Oz;  and  prolong  the  x  axis  beyond  0  to  x'.  Lay  off  on  Oythe 
value  of  n2;  on  Oz  the  value  of  r^;  z 

if  (n— n2)  is  positive,  lay  off  its 
value  n±  on  Ox  to  the  right;  if 
(n— n2)  is  negative,  lay  off  its  abso- 
lute value  n/=— (n  — n2)  on  Ox'' 
these  three  coordinates  will  deter- 
mine a  point  representing  a  solu- 
tion of  known  composition. 

If  this  point  M  is  to  the  right  £! 2; 

of  the  plane  yOz,  it  will  correspond 
to  a  positive  value  nt  of  (nl— n2); 
it  will  represent  a  solution  ob- 
tained by  mixing  with  18  grammes 
of  water  n^  grammes  of  potas-  .  FIG.  30. 

sium   sulphate,    n2^2   grammes   of   potassium   chloride,  and 
grammes  of  magnesium  sulphate. 

If  this  point  M'  is  to  the  left  of  the  plane  yOz,  it  will  correspond 
to  a  negative  value  — n/  of  (n— n2);  it  will  represent  a  solution 
obtained  by  mixing  with  18  grammes  of  water  W/TT/  grammes  of 
potassium  sulphate,  n27T2  grammes  of  potassium  chloride,  and  n^3 
grammes  of  magnesium  sulphate. 

This  is  the  method  of  representation  devised  by  Van't  Hoff. 

It  is  evident  that  what  was  discussed  in  Art.  113  may  be  treated 
from  this  standpoint;  under  a  given  pressure,  at  a  given  tempera- 
ture, each  of  the  salts  which  may  be  precipitated  corresponds  to 
a  surface  of  solubility. 

The  solid  bodies  which  the  preceding  solution  may  deposit  at 
25°  are  seven  in  number: 

Two  anhydrous  salts:  Potassium  chloride,  KC1 
Potassium  sulphate,  KgSOi 

Three  hydrated  salts :  MgS04  •  7H2ti 
MgS04-6H20 
MgCl2-6H2O 

Two  double  salts :         Schoenite,  igVIg(S04)2  •  6H20 

Carnallite,  KgKCL,  -  6H30 


138 


THERMODYNAMICS  AND  CHEMISTRY. 


It  seems,  besides,  that  three  other  substances  should  be  obtained: 


Two  hydra  ted  salts :  MgSO4  •  5H2O 
MgSO4-4H2O 
A  double  salt:  Leonite, 


K2Mg(S04)2-4H20 


In  any  case  the  conditions  of  the  formation  of  these  substances 
are  still  little  known.  The  authors  that  we  have  cited  limited 
themselves  to  the  study  of  the  first  seven;  they  have  constructed 
the  polyhedral  surface  of  seven  curved  faces  which  constitute  at 

n,(MgS04) 


(K2SOJ 


tt2(K2C!2) 

FIG.  31. 

25°  the  domains  of  these  salts.    Fig.  31  gives  a  general  idea  of 
this  surface. 

115.  System:  Water,  potassium  chloride,  sodium  chloride, 
potassium  sulphate,  sodium  sulphate.  Studies  of  Meyerhoffer 
and  Saunders. — For  each  temperature  there  corresponds  a  surface 
analogous  to  the  preceding;  for  the  system  just  studied  only  the 
surface  for  25°  has  been  constructed.  A  more  complete  study 
has  been  made  by  MeyerhofTer  and  Saunders  *  of  the  system 


H2O,    KC1,    Na2S04,    Nad, 


and  of  the  system 


H30,    KC1,    Na2S04,    K2S04 


1 W.  MEYERHOFFER  and  A.  P.  SAUNDERS,  Zeitschrift  fur  physikalische 
Chemie,  v.  28,  p.  453,  1899. 


MULTIVARIANT  SYSTEMS. 


139 


joined  by  the  relation 

2KC1 + Na2S04 = 2NaCl  +  K2S04. 

Regarding  these  systems  almost  exactly  the  same  words  may 
be  said  as  in  the  preceding  article,  replacing  Mg  by  Na^. 

Six  distinct  precipitates  may  be  formed  at  the  temperatures 
studied  by  Meyerhoffer  and  Saunders,  as  follows: 

1°.  Four  anhydrous  salts:  Sodium  chloride,       NaCl 

Potassium  chloride,  KC1 
Potassium  sulphate,  K2S04 
Sodium  sulphate,      Na2SO4 
Glauber's  salt,  Na^SO*  •  10H2O 

Glaserite,  K3Na(SO4)2 


2°.  An  hydrated  salt: 
3°.  A  double  salt: 


At  each  temperature  and  pressure  the  domains  of  the  six  salts 
form  a  polyhedron  of  six  curved  faces ;  Meyerhoffer  and  Saunders  l 
have  constructed  four  of  these  polyhedrons,  those  corresponding 
to  atmospheric  pressure  and  to  temperatures  0°,  4°.4,  16°. 3,  and 
25°. 

We  represent  here  (Figs.  32  and  33)  two  of  these  surfaces; 
Fig.  32  is  the  surface  for  0°,  and  Fig.  33  the  surface  for  25°. 


tt2   (K2C12) 
FIG.  32. 

It  is  to  be  noticed  that  Fig.  32  has  but  five  faces;  at  this  tem- 
perature the  anhydrous  sodium  sulphate  is  in  no  case  precipitated; 
no  domain  corresponds  to  this  salt.  The  same  is  true  for  the 

1 W.  MEYERHOFFER  and  A.  P.  SAUNDERS,  Zeitschrift  fur  physikalische 
Chemie,  v.  28,  p.  453,  1899. 


140 


THERMODYNAMICS  AND  CHEMISTRY. 


temperatures  4°.4  and  16°.3,  which  again  correspond  to  five  face 
surfaces.  On  the  contrary,  the  surface  for  25°  has  six  faces;  the 
anhydrous  sodium  sulphate  domain  figures  there  and  is  already 
of  considerable  size. 

116.  Four  salts  dissolved  in  water,  one  of  them  to  saturation. 
System:  Water,  sodium  chloride,  potassium  chloride,  sodium 
sulphate,  magnesium  chloride. — Suppose  that  into  the  system 
water,  potassium  chloride,  sodium  sulphate,  magnesium  chloride, 
we  introduce  a  new  independent  component,  which  we  shall  denote 


n3(Na2SO4) 


(K2C12) 

FIG.  33. 

by  the  index  4;  for  example,  sodium  chloride,  NaCl.  We  shall 
keep  the  index  1  for  potassium  chloride,  2  for  sodium  sulphate, 
3  for  magnesium  chloride.  The  system  is  then  formed  of  five 
independent  components,  c=5;  if  it  existed  in  only  two  phases, 
<£  =  2,  the  variance  F=c+2  —  <f>  would  be  equal  to  5;  this  quinti- 
variant  system  would  be  more  difficult  of  study  than  the  systems 
of  which  we  have  just  treated.  But  Van't  Hoff  and  his  pupils,1 

1 J.  H.  VAN'T  HOFF  and  A.  P  SAUNDERS,  Sitzungsberichte  der  Berliner 
Akademie,  1898,  p.  387;  VAN'T  HOFF  and  T.  ESTREICHER-ROZBIERSKI,  ibid., 
1898,  p.  487;  VAN'T  HOFF  and  MEYERHOFFER,  ibid.,  1898,  p.  590;  VAN'T 
HOFF,  Report  to  Physics  Congress  (Paris,  1900),  v.  i,  p.  464;  VAN'T  HOFF  and 
H.  v.  EULER  CHELPIN,  Sitzungsber.  Berl.  Akad.,  1900,  p.  1018;  VAN'T  HOFF 
and  MEYERHOFFER,  ibid.,  1901. 


MULTIVARIANT  SYSTEMS.  141 

not  satisfied  with  fixing  the  pressure  (1  atmosphere)  and  the  tem- 
perature (25°),  impose  upon  the  solution  the  condition  of  being 
constantly  saturated  with  sodium  chloride;  this  condition  was 
assuredly  fulfilled  in  the  circumstances  under  which  were  formed 
the  salts  deposits  of  Stassfurt.  They  then  seek  under  what  con- 
ditions this  solution  may  be  in  equilibrium  with  another  solid 
salt. 

In  other  terms,  they  study  this  solution,  formed  of  five  inde- 
pendent components,  in  the  presence  of  two  solid  phases,  one  of  which 
is  always  sodium  chloride. 

Let  C  be  the  second  solid  phase. 

Let  «4  be  the  ratio  of  the  mass  M4  of  sodium  chloride  contained 
in  the  solution  to  the  mass  M0  of  water  that  it  contains. 

The  pressure  n  and  the  temperature  T  having  invariable 
values,  every  time  the  two  concentrations  s1;  s3  of  the  potassium 
chloride  and  the  magnesium  chloride  in  the  solution  are  given 
arbitrarily,  we  shall  know  the  concentration  s2  of  the  sodium 
sulphate  and  the  concentration  s4  of  the  chloride  of  sodium 
within  a  solution  capable  of  resting  in  equilibrium  in  contact 
with  an  excess  of  sodium  chloride  and  a  solid  deposit  of  the 
salt  C.  In  order  to  show  completely  the  state  of  such  a  solution, 
one  might  make  use  simultaneously  of  two  representative  points: 
the  one,  M,  would  have  for  coordinates  the  three  concentrations  slf 
s2,  s3;  the  other,  //,  would  have  the  coordinates  slf  s3,  s4.  When 
the  two  concentrations  sl  and  s3  were  varied,  the  first  would 
describe  a  well-determined  surface  S  and  the  other  an  equally 
definite  surface  I.  The  simultaneous  knowledge  of  these  two 
surfaces  would  give  complete  information  on  the  subject  of  the 
solutions  which  may  remain  in  equilibrium,  under  atmospheric 
pressure  and  at  the  temperature  of  25°,  in  contact  with  an  excess 
of  common  salt  and  crystals  of  the  salt  C. 

Each  of  these  surfaces  would  possess  properties  similar  to  those 
possessed  by  the  unique  surface  S,  in  the  case  where  a  solution 
formed  of  four  independent  components  was  in  equilibrium  with 
a  single  solid  deposit. 

Suppose  that,  besides  sodium  chloride,  the  solution  may  pre- 
cipitate various  solid  salts  C,  C",  C".  .  . ;  to  each  of  these  salts 


142  THERMODYNAMICS  AND  CHEMISTRY. 

would  correspond  one  of  the  domains  S,  Sf,  S"  .  .  .  in  the  system 
of  coordinates  slt  s2.  s3,  and  also  one  of  the  domains  I,  2'  ,  2"  .  .  . 
in  the  system  slf  s3,  s4. 

In  Van't  Hoff's  researches  it  was  not  necessary  to  know  the 
concentration  s4  of  the  solution  in  sodium  chloride;  it  sufficed 
to  determine  the  composition  of  the  solution  which  rested  if  sodium 
chloride  was  precipitated. 

To  determine  this  composition  it  is  enough  to  know  the  re- 
spective concentrations  sl}  s2,  s3  of  potassium  chloride,  of  sodium 
sulphate,  and  of  magnesium  chloride;  but  there  may  be  substi- 
tuted for  these  thiee  quantities  three  others  which  will  give 
equally  well  the  composition  of  the  solution  after  all  the  sodium 
chloride  has  been  precipitated. 

Let  wlf  w2,  ws  be  the  molecular  weights  of  potassium  chloride, 
of  sodium  sulphate,  and  of  magnesium  chloride,  that  is  to  say, 
of  the  numbers  of  grammes  represented  by  the  formulae 

KC1,    Na2S04,    MgCl2. 

For  a  molecule  or  18  grammes  of  water  the  solution  to  analyze 
contains  n^  grammes  of  potassium  chloride,  n2w2  grammes  of 
sodium  sulphate,  and  n3w3  grammes  of  magnesium  chloride. 
Analysis  gives  the  numbers  nlt  n2)  n3,  and  as  evidently 


_n2w2         _ 
*~~~18'      3"  18  ; 


we  see  that  the  knowledge  of  the  numbers  nlt  n,,  n3  is  equivalent 
to  knowing  the  three  concentrations  s,,  s,,  s3. 

Instead  of  having  given  the  numbers  n»  n2,  na,  one  may,  if  it 
is  preferred,  determine  the  composition  of  the  solution  from  the 
numbers 


MULTIVARIANT  SYSTEMS. 


143 


These  are  the  three  quantities  x,  y,  and  z  that  Van't  Hoff  lays 
off  on  the  three  axes  of  rectangular 
coordinates  (Fig.  34). 

Each  point  represents  then  a  solu- 
tion which  would  have  a  well-deter- 
mined composition  after  having  pre- 
cipitated the  sodium  chloride  which 
saturates  it. 

At  a  given  temperature  and  under 
a  given  pressure  each  of  the  precipi- 
tates, other  than  sodium  chloride, 
which  may  be  formed  in  the  system 
corresponds  to  a  domain. 

There  are  fourteen  substances, 
always  present  in  excess,  which  may 
be  precipitated  with  the  sodium  chlo- 
ride at  25°  C.  and  atmospheric  pres- 
sure: 

(1)  KC1 

(2)  Na2S04 


X  (KCl) 


Two  anhydrous  salts: 
Six  hy drated  salts : 


FIG.  34. 


(3)  MgCV6H2O 

(4)  4MgSO4-5H2O 

(5)  MgSO4-4H2O 

(6)  MgSO4-5H2O 

(7)  MgS04-6H20 

(8)  MgS04-7H,O 

Six  double  salts :  (9)  MgKCL,  •  6H2O  (Carnallite) 

(10)  MgK2(SO4)2-4H20(Leonrte) 

(11)  MgK,(SO4)2-6H2O(Schoenite) 

(12)  MgNa2(SO4)2'4H20(Astrakanite) 

(13)  K3Na(SO4)2  (Glaserite) 

(14)  MgSO4-KCl-3H2O(Kainite) 

Fig.  34  represents  the  polyhedron  of  13  curved  faces  that  the 
domains  of  these  salts  form.  The  numbers  preceding  the  salts 
in  the  column  correspond  in  Fig.  34  to  their  respective  domains. 

117.  Five  salts  dissolved  in  water,  two  of  which  to  saturation; 
a  calcium  salt  added  to  the  preceding  system. — A  sixth  com- 


144  THERMODYNAMICS  AND  CHEMISTRY. 

ponent  may  be  added  to  the  preceding  solution;  provided  that  the 
number  of  precipitates  in  contact  with  which  this  solution  re- 
mains in  equilibrium  be  also  increased  by  unity,  the  variance  of 
the  system  will  keep  the  same  value. 

Suppose  that  to  the  sodium  chloride,  potassium  chloride, 
sodium  sulphate,  magnesium  chloride  we  add  calcium  sulphate 
to  which  we  give  the  index  5.  Five  concentrations,  s1;  S2,  s3,  s4,  s5, 
will  determine  the  composition  of  the  solution. 

The  calcium  salts  being  very  slightly  soluble,  one  of  them  will 
be  precipitated;  but  depending  upon  the  conditions  of  composi- 
tion of  the  solution,  temperature,  and  pressure  the  calcium  salt 
precipitated  will  be  different. 

It  is  question,  then,  of  studying  the  solutions  which  may  re- 
main in  equilibrium  in  contact  with  three  precipitates  one  of  which 
will  always  be  sodium  chloride  and  another  always  a  calcium  salt 
ft  the  third  precipitate,  C,  will  be  a  salt  of  sodium,  magnesium, 
or  again  a  double  salt  containing  two  of  these  metals. 

Suppose  given  the  temperature,  25°,  and  the  pressure,  one 
atmosphere.  If  one  wishes  the  solution  to  remain  in  equilibrium 
in  contact  with  common  salt  and  a  couple  of  precipitates  (C,  f), 
there  may  be  given  arbitrarily  the  values  of  two  of  the  five  con- 
centrations slf  s2,  s3,  s4,  s5,  and  the  values  of  the  other  three  will 
follow.  We  see  then  that  at  a  given  pressure  and  temperature 
each  couple  of  precipitates  (C,  7-),  of  which  the  first  does  not  contain 
lime  and  the  second  is  a  calcium  salt,  will  correspond  to  a  certain 
domain  S  in  the  system  of  coordinates  s1}  s2,  s3;  to  another  domain 
2  in  the  system  sv  s3,  s4;  finally,  to  a  third  domain  in  the  system 

o       o       o  •* 

*l;   *3>   *V 

It  is  by  means  of  the  first  coordinate  system,  or,  rather,  by 
means  of  the  equivalent  coordinate  system  x,  y,  z,  defined  in  the 
preceding  article,  that  Van't  Hoff l  and  his  pupils  have  repre- 
sented the  domains  of  various  pairs  of  two  salts  which  may  be 
precipitated  in  the  midst  of  the  system. 

This  case  may  be  simplified.  The  calcium  salts  being  very 
slightly  soluble,  the  concentration  s5  is  always  near  to  0.  It  fol- 
lows that  the  values  which  the  concentrations  slf  s2,  s3,  s4  ought 

1  VAN'T  HOFF,  Reports  to  Physics  Congress  (Paris,  1900),  v.  I,  p.  464. 


MULTIVARIANT  SYSTEMS. 


145 


to  have  in  order  that  the  solution  be  in  equilibrium  in  contact 
wi'th  common  salt  and  with  the  couple  (C,  7-)  are  sensibly  the 
same  as  if  the  concentration  s5  was  equal  to  0  and  the  calcium 
salt  f  suppressed.  For  the  coordinate  system  sl}  s2,  s3;  or  what 
amounts  to  the  same,  for  the  system  x,  y,  z,  the  domain  of  the 
couple  (C,  7-)  should  almost  exactly  coincide  with  the  domain 
found  for  the  salt  C  in  the  preceding  article,  or  with  a  part  of  this 
domain. 

One  may  therefore  begin  by  determining  the  domains  of  the 
salts  C,  C'y  C"  ...  as  if  the  system  did  not  contain  calcium  sul- 
phate, and  this  will  give  the  surface  represented  by  Fig.  34.  Next 
it  should  be  seen  if,  throughout  the  extent  of  one  of  these  domains 
S,  the  salt  C  is  associated  with  the  same  calcium  salt  ;*,  or  if,  on 
the  contrary,  it  is  associated  with  various  calcium  salts  7^,  f21 .  .  . , 
case  in  which  the  domain  S  should  be  subdivided  into  various 
subdomains  Slt  S2, .  .  . ,  corresponding  respectively  to  the  couples 

(C,  n),  (C,  r»),  •  •  • 

The  result  of  this  discussion  is  represented  by  Fig.  35.  The 
solid  lines  map  the  surface  which  has 
already  been  represented  in  Fig.  34. 
The  dotted  lines  divide  the  domains  of 
the  various  salts  of  potassium,  magne- 
sium, or  sodium  into  subdomains  rela- 
tive to  the  various  calcium  salts  which 
may  be  precipitated  in  the  conditions 
of  the  experiments. 

These    calcium    salts    are    four    in 
number: 

2CaSO4-H2O 
CaSO4-2H2O  (Gypsum) 
CaNa2(SO4)2  (Glauberite) 
CaKjCSOJa-HO  (Syngenite) 

The  subdomains  corresponding  to 
the  semi-hydrated  calcium  sulphate  are 
bounded  by  the  line  ABODE;  the  sub- 
domains  for  gypsum  are  bounded  by 
the  line  CDEIHGFN;  glauberite  precipitates  in  the  region  IHJKP, 
and  the  syngenite  in  the  region  NFGHJKLM. 


146 


THERMODYNAMICS  AND  CHEMISTRY. 


We  see,  therefore,  that  the  following  associations  in  the  presence 
of  common  salt  in  excess  may  be  observed  between  a  calcium  salt 
and  a  non-calcium  salt: 


MgC!2-6H20 
4MgS04-5H2O 
MgSO4-4H2O 
MgS04-5H,0 
Carnallite 
MgS04-6H20 
KC1 

MgS04-7H2O 
Astrakanite  ) 
Na2S05         [ 
Kainite 
Leonite     ) 
Schoenite  >• 
Glaserite  ) 


with2CaSO4-H2O; 

with  2CaS04-H2O  or  Gypsum; 

with  2CaS04-H2O,  Gypsum  or  Syngenite; 

with  Gypsum  or  Syngenite; 

with  Gypsum,  Syngenite,  or  Glauberite; 

with  Syngenite  or  Glauberite ; 
with  Gypsum  or  Syngenite ; 

with  Syngenite. 


These  results  refer,  it  is  understood,  to  the  temperature  of  25°. 

They  show  how  the  phase  rule  guides  the  experimentalist  in 
the  analysis  of  systems  whose  complexity  would  otherwise  defy 
all  attacks. 


CHAPTER  VIII. 
MONOVARIANT  SYSTEMS. 

118.  Return  to  monovariant  systems.  —  After  having  shown, 
by  several  examples,  the  services  that  the  phase  rule  may  render 
in  the  discussion  of  complicated  cases  occurring  with  trivariant 
or  quadrivariant  systems,  we  return  to  study  more  in  detail  the 
properties  of  systems  whose  variance  has  a  smaller  value,  com- 
mencing with  monovariant  systems. 

We  call  thus  every  system  divided  into  <j>  phases  related  to  the 
number  c  of  components  by  the  relation 


For  such  a  system  the  variance  V=c+2  —  cj>  is  equal  to  1. 

119.  One  component  existing  in  two  phases.  —  Among  the 
monovariant  systems  we  find  in  the  first  place  all  those  where 
a  single  component  is  divided  between  two  phases;  among  the  most 
remarkable  examples  are: 

1°.  A  solid  or  liquid  substance  is  in  the  presence  of  its  own 
vapor;  the  study  of  such  systems  constitutes  the  theory  of  the 
vaporization  of  liquid  or  solid  bodies. 

2°.  A  single  substance  exists  simultaneously  in  the  two  states, 
liquid  and  solid,  case  to  which  applies  the  theory  of  the  fusion  of 
solids  and  the  congelation  of  liquids. 

3°.  A  single  chemical  substance,  simple  or  compound,  exists 
at  once  in  two  different  solid  forms,  such  as  the  yellow  iodide  of 
mercury  with  the  red  iodide. 

4°.  A  gaseous  chemical  substance  exists  in  the  presence  of 
a  solid  polymer,  as  gaseous  cyanogen  in  the  presence  of  solid  para- 
cyanogen,  or,  again,  gaseous  cyanic  acid  with  crystallized  cyanuric 
acid. 

147 


148  THERMODYNAMICS  AND  CHEMISTRY. 

1 20.  Two  components  divided  among  three  phases. — We  find 
next,  among  the  mono  variant  systems,  those  formed  of  two  inde- 
pendent components  divided  into  three  phases;    among  them  let  us 
mention :    . 

1°.  The  systems  where  a  solid  component  and  a  gaseous  com- 
ponent exist  in  the  presence  of  a  solid  compound;  such  is  the 
system  formed  by  lime,  carbonic  acid  gas,  calcium  carbonate,  or, 
again,  the  system  consisting  of  an  anhydrous  salt,  water  vapor, 
a  definite  hydrate  of  the  same  salt. 

2°.  The  systems  in  which  two  independent  components,  water 
and  an  anhydrous  salt,  exist  in  three  phases:  a  solid  precipitate, 
anhydrous  or  hydrated,  a  solution,  and  water  vapor. 

3°.  The  systems  where  a  solid  hydrate  of  a  gas  is  in  contact 
with  an  aqueous  solution  of  this  gas  and  a  mixture  of  the  gas  with 
water  vapor. 

4°.  A  mixture  of  two  liquids,  separated  into  two  layers,  be- 
neath a  simple  or  mixed  vapor,  formed  from  one  or  both  of  these 
liquids. 

121.  Three  components   with   four  phases. — A    system    for 
which  three  independent  components  are  divided  among  four  phases 
is  also  a  monovariant  system.     As,  for  example: 

The  three  independent  components  are  water,  0,  and  two  an- 
hydrous salts,  1  and  2;  the  four  phases  are  water  vapor,  an  aqueous 
solution  of  the  two  salts,  and  two  solid  precipitates,  C,  C",  which 
are  bodies  of  definite  composition  formed  from  the  three  inde- 
pendent components  0,  1,  2  (icer  anhydrous  salts,  hydrated  salts, 
double  anhydrous  or  hydrated  salts). 

122.  Law  of  equilibrium  for  monovariant  systems.     Trans- 
formation tension  and  transformation  point.— The    equilibrium 
states  of  a  monovariant  system  obey  a  law  which,  for  all  these 
systems,  has  the  same  form,  as  follows: 

At  a  given  temperature  the  pressure  for  which  the  system  is 
in  equilibrium  has  an  entirely  definite  value,  which  is  called  the 
TRANSFORMATION  TENSION  at  the  temperature  considered.  The 
composition  and  density  of  each  phase  which  composes  the  sys- 
tem in  equilibrium  are  equally  definite;  like  the  transformation 
tension,  they  do  not  depend  upon  the  masses  of  the  independent 
components  which  constitute  the  system.  On  the  contrary,  the 


MONOVARIANT  SYSTEMS.  149 

masses  of  the  various  phases  are  not  entirely  determined,  even 
when  the  masses  of  the  independent  components  are  given. 

Under  a  given  pressure  the  temperature  for  which  the  system  is 
in  equilibrium  has  a  definite  value,  which  is  called  the  TRANSFORMA- 
TION POINT  under  the  pressure  considered;  this  equilibrium  tem- 
perature does  not  depend  upon  the  masses  of  the  independent 
components  which  make  up  the  system,  and  it  is  the  same  with 
the  composition  and  density  possessed,  at  the  moment  of  equi- 
librium, by  each  of  the  phases  into  which  the  system  is  divided; 
nevertheless  the  masses  of  these  phases  are  not  entirely  determined, 
even  where  the  masses  of  the  independent  components  are  known. 

123.  Curve  of  transformation  tensions. — Take  two  axes  of 
rectangular  coordinates,  OT,  On  (Fig.  36);  on  the  axis  of  abscissae 
OT  lay  off  a  length  OT  measured  by  thec 
temperature  that  we  consider;  through  the 
point  T  draw  a  parallel  to  the  axis  of  ab- 
scissse  On,  and  on  this  parallel  lay  off  a  length 
TM=OP  measured  by  the  transformation 
tension  at  the  temperature  considered;  when 
the  temperature  assumes  all  possible  values 


and  the  point  T  describes  the  line  OT,  the  ° 
point  M  describes  the  curve  CC',  called  the  Ro-  36- 

curve  of  transformation  tensions  of  the  monovariant  system  con- 
sidered. 

Suppose  the  curve  of  transformation  tensions  traced;  if  we 
know  a  temperature  T=OT,  a  simple  construction  will  give  us 
the  corresponding  transformation  tension  P—OP\  this  will  be 
the  ordinate  of  the  point  M  on  the  curve  CC',  which  has  OT  as 
abscissa;  if  there  is  given  a  pressure  P=OP,  a  simple  construc- 
tion gives  us  the  corresponding  transformation  temperature  T=OT; 
this  is  the  abscissa  of  the  point  M  on  the  curve  CC'  which  has  the 
ordinate  OP. 

124.  Curve  of  tensions  for  saturated  vapor. — It  was  in  the 
study  of  the  vaporization  of  solid  and  liquid  bodies  that  physi- 
cists met,  for  the  first  time,  the  curve  of  transformation  tensions 
of  a  monovariant   system.     The   curve,  in  this  case,  is  nothing 
else  than  the  curve  of  tensions  for  saturated  vapor. 

125.  Fusion  phenomena.— The  existence  of  a  curve  of  trans- 


150  THERMODYNAMICS  AND  CHEMISTRY. 

formation  tensions  in  the  phenomena  of  fusion  was  predicted 
theoretically  in  1849  by  James  Thomson  and  verified  experimentally 
in  1850  by  William  Thomson;  these  physicists  showed  that  the 
fusing-point  of  ice  has  not  an  absolutely  fixed  value,  but  changes 
when  the  pressure  is  changed  which  acts  upon  the  mono  variant 
system  composed  of  water  and  ice;  other  physicists  have  since 
established  the  same  truth  in  studying  the  fusion  of  other  sub- 
stances; but,  for  reasons  that  we  shall  see  later,  it  is  necessary  to 
cause  the  pressure  to  vary  greatly  in  order  to  produce  an  appre- 
ciable change  in  a  fusing-point;  the  curve  of  transformation  ten- 
sions deviates  only  slightly  from  a  straight  line  parallel  to  On. 

126.  Allotropic   transformations    of   solids. — As    much    may 
be  said  of  the  curve  for    transformation    tensions    in    the    case 
where    one    substance   may  exist   in   two   allotropic   forms,  both 
being  solid;   nevertheless,  if  the  pressure  upon  the  system  under- 
goes great  changes,  notable  variations  of  the  transformation  point 
may  be  obtained;  thus  the  red  silver  iodide  may  be  transformed 
into  the  yellow  silver  iodide;   it  suffices  for  this,  at  atmospheric 
pressure,  to  increase  the  temperature  to  +146°  C.;  by  raising  the 
pressure  to  some   3000  atmospheres  Mallard  and  Le  Chatelier  l 
were  able  to  lower  the  transformation  point  to  ordinary  tempera- 
tures. 

127.  Gaseous  substance  and  solid  polymer. — The   curve   of 
transformation  tensions  in  a  system  where  a  gaseous  body  exists 
in  presence  of  a  solid  polymer  has  an  appearance  similar  to  that 
of  a  curve  for  the  tension  of  saturated  vapor;  this  curve  rises  from 
left  to  right,  and  that  the  more  steeply  as  the  temperature  is  higher; 
the  existence  of  such  a  curve  was  discovered  first  by  Troost  and 
Hautefeuille 2  for  the  systems  where  gaseous  cyanogen  exists  in 
presence  of  solid  paracyanogen,  then  for  systems  where  the  gase- 
ous cyanic  acid  exists  with  solid  cyanuric  acid. 

128.  Dissociation  tensions.— The  equilibrium  of  a  system  in 
which  a  definite  solid  component  exists  with   two   independent 
components,  one  of  which  is  solid,  the  other  gaseous,  necessitates 

1  MALLARD  and    LE  CHATELIER,  Journal   de  Physique,  2d  Series,  v.  4, 
p.  305,  1885. 

2  TROOST  and  HAUTEFEUILLE,  Annales  de  I'Ecole  normale  superieure,  2d 
Series,  v,  2,  p.  253,  1873. 


MONOVARIANT  SYSTEMS.  151 

that  at  each  temperature  the  pressure  supported  by  the  system  be 
equal  to  the  transformation  tension  for  the  temperature  considered; 
this  is  the  fundamental  law  demonstrated  by  H.  Debray,  first l  in 
studying  the  dissociation  of  calcium  carbonate  into  lime  and  car- 
bonic acid  gas,  then 2  in  studying  the  dissociation  of  certain  hy- 
drated  salts  into  anhydrous  salts  and  water  vapor. 

Put  with  certain  metallic  chlorides  ammonia  gas  is  absorbed 
by  these  chlorides  and  forms  with  them  definite  solid  compounds; 
the  dissociation  of  an  ammonia  chloride  into  a  metallic  chloride 
and  ammonia  gas  corresponds  to  a  curve  of  transformation  ten- 
sions which  is  here  called  the  curve  of  dissociation  tensions.  Isam- 
bert 3  has  determined  a  certain  number  of  these  curves  and  has 
shown  their  analogy  to  the  curve  fo:  vapor  tension  of  saturated 
vapors  from  liquids;  his  work  has  since  been  completed  by  Joan- 
nis  and  Croizier.4 

Other  curves  of  dissociation  tension  have  been  determined  by 
chemists;  we  only  note  here  the  curves,  determined  by  Joannis,5 
of  the  tensions  of  dissociation  of  potassammonium  into  potassium 
and  ammonia  gas,  and  of  sodammonium  into  sodium  and  ammonia. 

A  system  which  encloses,  simultaneously,  an  aqueous  solution 
of  a  gas,  a  mixture  of  this  gas  with  water  vapor,  and  a  definite 
solid  compound  formed  by  the  union  of  the  gas  and  water  is  in 
equilibrium,  at  each  temperature,  when  the  pressure  has  a  defi- 
nite value;  the  liquid  mixture  and  the  gaseous  mixture  has,  at 
the  same  time,  a  definite  composition;  the  total  mass  of  gas  and 
the  total  mass  of  water  contained  in  the  system  do  not  influence 
either  this  tension  nor  this  composition;  this  law  was  first  recog- 
nized by  Isambert 6  in  studying  the  dissociation  of  chlorine  hy- 
drate; the  curves  of  transformation  tension  of  a  great  number 


1  H.  DEBRAY,  Comptes  Rendus,  v.  64,  p.  603,  1867. 

2  Ibid.,  v.  66,  p.  194,  1868. 

*  ISAMBERT,  Comptes  Rendus,  v.  66,  p.  1259,  1868;  Ann.  d.  I'Ecole  normal 
*up.,  v.  5,  p.  129,  1868. 

4  JOANNIS  and  CROIZIER,  Memoires  de  la  Societe  des  sciences  physiques 
et  naturelles  de  Bordeaux,  4th  Series,  v.  5,  p.  41,  1895. 

8  JOANNIS,  ibid,,  v.  5,  p.  218,  1895. 

8  ISAMBERT,  Comptes  Rendus,  v.  86,  p.  481,  1898. 


152  THERMODYNAMICS  AND  CHEMISTRY. 

of  analogous  systems  have  been  determined  by  H.  Le  Cha teller, 
Wroblewski,  Bakhuis  Roozboom,  and  P.  Villard.1 

129.  Transformation  point  of  double  salts. — Consider  a  sys- 
tem formed  by  three  independent  components,  as 

Water:  H20; 

Copper  acetate:    Cu(C2H302)2; 
Calcium  acetate:  Ca(C3H3O2)2. 

This  system  may  be  divided  into  four  phases,  thus: 
A  liquid  mixture  containing  the  three  independent  components: 
Crystals  of  hydra  ted  copper  acetate:  Cu(C2H3O2)2-H.,O; 
Crystals  of  hydrated  calcium  acetate:  Ca(C2H3O2)2-H2O; 
Crystals    of   a   hydrated    double    salt,    Cupricalcium   acetate: 

CuCa(C2H302)4-6H20. 

Such  a  system  is  monovariant;  under  a  given  pressure  there 
exists  a  well-determined  temperature  for  which  it  may  exist  in 
equilibrium;  this  temperature  is  the  transformation  point  of  the 
cupricalcium  acetate  under  the  given  pressure;  if  the  tempera- 
ture is  less  than  the  transformation  point,  the  two  simple  salts 
combine  and  the  cupricalcium  acetate  is  formed;  if,  on  the  con- 
trary, the  temperature  is  higher  than  the  transformation  point, 
the  cupricalcium  acetate  decomposes  and  there  is  formed  copper 
acetate  and  calcium  acetate;  calcium  acetate  is  un colored,  copper 
acetate  green,  and  cupricalcium  acetate  blue,  so  that  these  trans- 
formations are  accompanied  by  color  changes  which  facilitate  the 
study,  as  was  first  shown  by  Van't  Hoff  and  Ch.  van  Deventer.2 
Reicher  3  has  shown  that  the  transformation  point  of  cupricalcium 
acetate,  at  atmospheric  pressure,  is  included  between  +76°  C. 
and  78°  C. 

This  transformation  point  should  depend  upon  the  pressure 
acting  upon  the  system;  but  here,  as  in  the  fusion  phenomena, 
it  is  necessary  to  cause  the  pressure  to  vary  greatly  in  order 
to  produce  an  appreciable  change  in  the  transformation  point; 

1  Memoir  by  P.  VILLARD  (Annales  de  Chimie  et  de  Physique,  7th  Series, 
v.  ii f  p.  289,  1897)  contains  a  complete  bibliography  of  the  subject. 

2  VAN'T  HOFF  and  VAN  DEVENTER,  Recueil  des  Travaux  chimiques  des  Pays 
Bos,  v.  6,  p.  407,  1886;  Zeitschnft  fur  physikalische  Chemie,  v.  I,  p.  163,  1887. 

8  REICHER,  Zeit.  phys.  Chem.,  v.  i,  p.  221,  1887. 


MONOVARIANT  SYSTEMS.  153 

W.  Spring  and  Van't  Hoff l  have  s  eenthis  point  lowered  to  +40°  C. 
under  a  pressure  of  about  6000  atmospheres. 

130.  Precautions  to  take  in  the  application  of  the  preceding 
laws.  First  example :  dissociation  of  the  red  oxide  of  mercury. 
Pelabon's  investigations.— The  application  to  a  chemical  system 
of  the  idea  of  the  curve  of  transformation  tensions  may,  in  certain 
cases,  necessitate  certain  precautions,  which  if  neglected  lead  to 
errors. 

For  example,  it  may  happen  that  with  the  same  chemical  sub- 
stances different  mono  variant  systems  may  be  constituted;  to 
each  one  of  these  monovariant  systems  will  correspond  a  curve  of 
transformation  tensions,  but  these  curves  will  not  be  superposable. 

The  dissociation  of  the  oxide  of  mercury  offers  an  interesting 
example  to  which  this  remark  applies. 

Suppose  that  the  red  oxide  decomposes  into  oxygen  and  mer- 
cury vapor  in  an  enclosure  empty  at  the  start.  The  system  is 
divided  into  two  phases:  solid  mercuric  oxide  and  the  gaseous 
mixture  of  oxygen  and  mercury  vapor.  How  many  independent 
components  does  it  contain?  In  the  conditions  which  we  suppose 
realized  it  is  sufficient  to  know  the  total  mass  of  mercury,  free 
or  combined,  that  it  contains.  The  system  therefore  does  not  con- 
tain two  independent  components,  but  merely  one,  oxide  of  mer- 
cury; it  consists  of  an  arbitrary  mass  of  mercuric  oxide,  partly 
in  a  state  of  combination,  partly  decomposed.  Summing  up,  we 
may  say  that  we  are  dealing  with  a  system  formed  of  a  single 
independent  component,  the  substance  HgO,  wrhich  exists  in  two 
phases,  a  solid  phase,  the  red  oxide,  and  a  gaseous  phase,  the 
mixture  in  equivalent  proportions  of  oxygen  and  mercury  vapor. 
Such  a  system  is  monovariant;  it  admits  a  curve  C  of  transforma- 
tion tensions;  at  each  temperature  T  the  curve  C  shows  a  corre- 
sponding transformation  tension  P. 

Instead  of  supposing  that  the  oxide  of  mercury  dissociates 
in  an  enclosure  empty  at  the  start,  it  may  be  supposed  to  dis- 
sociate into  an  enclosure  where  oxygen  or  mercury  vapor  has 
been  introduced;  in  this  case  the  mass  of  oxygen,  whether  free 
or  combined,  and  the  free  or  combined  mercury,  which  the  system 

1  SPRING  and  VAN'T  HOFF,  Zeit.  phys.  Chem.,  v.  i,  p.  227,  1887. 


154  THERMODYNAMICS  AND  CHEMISTRY. 

includes,  are  no  longer  in  equivalent  proportions;  these  two  masses 
are  arbitrary;  the  system  is  composed  of  two  independent  com- 
ponents, oxygen  and  mercury. 

If  the  system  consists  only  of  solid  mercuric  oxide  and  a  mix- 
ture of  oxygen  and  mercury  vapor,  it  is  divided  into  two  phases 
only  and  is  bivariant;  for  such  a  system  there  is  no  longer  ques- 
tion of  a  curve  of  transformation  tensions;  the  knowledge  of  the 
temperature  is  not  sufficient  to  determine  the  pressure  supported 
by  the  system  in  equilibrium. 

It  is  no  longer  the  same  if  there  is  introduced  into  the  system 
enough  mercury  so  that  a  part  of  this  substance  remains  in  the 
liquid  state;  the  system,  formed  of  two  independent  components, 
oxygen  and  mercury,  and  divided  into  three  phases,  red  oxide  of 
mercury,  mixture  of  oxygen  and  mercury  vapor,  and  liquid  mer- 
cury, is  a  mono  variant  system;  it  admits  of  a  curve  of  transfor- 
mation tensions  C";  at  each  temperature  T  the  curve  C"  has  a 
corresponding  transformation  tension  P'  whose  value  is  inde- 
pendent of  the  masses  of  mercury  and  oxygen  which  the  system 
contains . 

The  two  curves  have  been  determined  by  H.  Pelabon; 1  they 
are  in  nowise  identical;  the  curve  C'  is  much  higher  than  the. 
curve  C;  for  instance,  at  the  temperature  of  520°  C.  the  trans- 
formation tension  which  we  have  denoted  by  P  is  measured  by 
417.6  centimetres  of  mercury,  and  the  transformation  tension  de- 
noted by  P'  by  844.0  centimetres  of  mercury.  From  this  we  see 
how  unsafe  it  would  be  to  speak,  without  further  limitations,  of 
the  dissociation  tension  of  mercuric  oxide  at  a  given  temperature. 

131.  Second  example:  Dissociation  of  cupric  oxide. — Other 
precautions  should  be  taken  in  the  application  of  the  idea  of 
the  curve  of  transformation  tension  to  a  system;  it  may  happen 
that  a  phase  appears  or  disappears  in  the  system;  that  con- 
sequently the  system  is  monovariant  or  bivariant  according  to 
circumstances;  that  it  has  or  has  not  a  curve  of  transforma- 
tion tensions. 

Here  is  a  good  example  taken  from  the  investigations  of  De- 
bray  and  Joannis.2 

1  H.  PELABON,  Memoires  de  la  Societe  des  sciences  physiques  et  naturelles 
de  Bordeaux,  5th  Series,  v.  5,  1899. 

3  H.  DEBRAY  and  JOANNIS,  Comptes  Rendus,  v.  99,  pp.  583  and  688,  1884. 


MONOVARIANT  SYSTEMS.  155 

Cupric  oxide  dissociates  into  cuprous  oxide  and  oxygen.  Two 
independent  components,  cuprous  oxide  and  oxygen,  form  the 
system,  which]  below  a  certain  temperature  is  divided  among 
three  phases,  solid  cupric  oxide,  solid  cuprous  oxide,  oxygen  gas. 
The  system  is  monovariant,  admitting  a  curve  of  dissociation 
tensions. 

Thus,  at  a  given  temperature  T,  equilibrium  corresponds  to 
a  tension  P  of  the  oxygen  atmosphere,  perfectly  determined  by 
the  knowledge  of  the  single  temperature  T.  If  oxygen  is  intro- 
duced into  the  recipient  so  that  the  pressure  takes  on  momen- 
tarily a  value  greater  than  P,  the  cuprous  oxide  absorbs  oxygen 
and  is  transformed  into  cupric  oxide  until  the  pressure  has  again 
become  equal  to  P.  If,  on  the  contrary,  oxygen  is  removed, 
lowering  the  pressure  below  P,  cupric  oxide  is  reduced  until  the 
pressure  retakes  its  former  value. 

These  phenomena  are  very  sharply  produced  when  the  tem- 
perature does  not  exceed  a  certain  limit,  near  the  fusing-point  of 
gold;  beyond  this  limit  the  two  oxides  exist  in  contact  as  a  fused 
mass;  instead  of  forming  two  solid  phases,  they  form  but  one 
liquid  phase;  from  monovariant  the  system  becomes  bi variant; 
at  a  given  temperature  there  can  be  no  longer  any  definite  dis- 
sociation tension. 

Suppose,  for  example,  that  at  a  temperature  T  the  system 
is  in  equilibrium  and  the  liquid  mixture  of  cupric  and  cuprous 
oxides  in  the  presence  of  an  oxygen  atmosphere  under  the  pres- 
sure P;  without  changing  the  temperature,  let  us  introduce  a  cer- 
tain quantity  of  oxygen  so  as  to  increase  the  pressure;  the  liquid 
will  absorb  a  part,  but  only  a  part,  of  the  mass  introduced; 
when  equilibrium  is  again  established  the  pressure  will  have  a 
value  P'  higher  than  P;  if,  conversely,  we  had  withdrawn  oxygen 
so  as  to  have  lowered  the  pressure  to  less  than  the  initial  pressure 
P,  the  liquid  would  have  liberated  oxygen  so  as  to  increase  the 
pressure,  but  only  to  a  value  P'  less  than  P. 

132.  Dissociation  of  chlorine  hydrate. — Analogous  facts,  which 
caught  the  attention  of  Isambert  and  Le  Chatelier,  are  observed  in 
the  study  of  systems  having  for  independent  components  water 
and  chlorine;  at  low  temperatures  one  may  observe  such  a  system 
divided  into  three  phases:  chlorine  hydrate  crystals,  a  liquid 


156  THERMODYNAMICS  AND  CHEMISTRY. 

solution,  a  gaseous  atmosphere  of  chlorine  and  water  vapor;  at 
each  temperature  equilibrium  is  established  when  the  pressure  has 
a  value  which  depends  upon  the  temperature  alone  and  in  nowise 
upon  the  masses  of  chlorine  and  water  enclosed  by  the  system. 

If  the  temperature  attains,  then  exceeds,  the  point  of  aqueous 
fusion  of  these  crystals  of  chlorine  hydrate,  they  disappear;  the 
system  which  then  contains  but  two  phases,  the  liquid  mixture 
and  the  gaseous  mixture,  is  bivariant;  at  a  given  temperature  the 
tension  of  the  gaseous  mixture  which  remains  in  equilibrium  above 
the  liquid  solution  may  assume  an  infinity  of  values;  to  increase 
it,  the  addition  of  chlorine  is  sufficient;  to  decrease  it,  removal  of 
a  part  of  the  gaseous  mixture. 

133.  The  absence  of  a  fixed  tension  of  dissociation  distin- 
guishes a  solution  from  a  definite  compound. — These  remarks 
suggest  the  means  of  deciding,  in  certain  cases,  some  questions 
in  dispute. 

A  gas,  ammonia  for  example,  is  absorbed  by  a  solid  substance 
C;  what  is  the  nature  of  this  absorption?  Does  the  ammonia  gas 
form  with  the  solid  a  definite  compound  whose  particles  are  dis- 
seminated through  the  solid  C  in  excess,  but  nevertheless  distinct 
from  the  solid?  Or,  on  the  contrary,  does  it  form  with  the  solid  a 
solid  solution  of  which  each  infinitely  small  volume  contains  both 
matter  from  the  solid  and  from  the  ammonia? 

The  system  is  assuredly  formed  of  two  independent  compo- 
nents, ammonia  gas  and  the  solid  C. 

According  to  the  first  hypothesis  the  system  is  divided  into 
three  phases,  which  are  ammonia  gas,  the  solid  C,  the  solid  am- 
moniacal  compound;  the  system  is  therefore  monovariant;  for 
each  temperature  T  should  correspond  a  definite  dissociation  ten- 
sion P;  if  the  ammonia  gas  is  removed,  a  certain  quantity  of  the 
solid  ammoniacal  compound  will  be  dissociated  so  as  to  restore 
the  value  P  to  the  tension  of  the  gaseous  atmosphere,  and  this 
operation  may  be  repeated  several  times  up  to  the  moment  when 
the  whole  of  the  solid  ammoniacal  compound  is  destroyed;  if 
ammonia  gas  is  now  added,  the  excess  of  gas  will  be  absorbed  by 
the  solid  C  and  the  tension  brought  back  to  the  value  P  at  the 
moment  when  the  whole  of  the  substance  C  has  changed  over 
to  the  state  of  ammoniacal  compound. 


MONOVARIANT  SYSTEMS.  157 

Quite  otherwise  will  be  the  succession  of  phenomena  accord- 
ing to  the  second  hypothesis;  the  system,  formed  only  of  two 
phases,  ammonia  gas  and  the  solid  solution,  is  bivariant;  at  a 
single  temperature  T  the  system  may  be  observed  in  equilibrium 
for  an  infinity  of  different  values  for  the  tension  of  the  ammonia 
gas;  the  pressure  exerted  by  this  gas  at  the  moment  of  equilib- 
rium will  increase  if  ammonia  gas  is  added  to  the  system,  and  will 
diminish  if  gas  is  withdrawn. 

As  long  ago  as  1867,  Isambert  made  use  of  this  criterion  to 
demonstrate  that  ammonia  gas  when  absorbed  by  metallic  chlo- 
rides forms  with  these  bodies  definite  compounds,  while  when 
absorbed  by  carbon  it  forms  with  this  substance  a  solid  solution. 

134.  The  zeolites  are  solid  solutions. — In  the  last  few  years 
the  criterion  suggested  by  Isambert  has  been  used  anew  and  has 
furthered  the  establishment  of  interesting  consequences. 

Nature  furnishes  us  wiih  a  certain  number  of  hydrated  sili- 
cates which  mineralogists  call  zeolites;  the  dehydration  of  certain 
hydrates  offers  curious  peculiarities;  analcime,  for  instance,  may 
be  completely  dehydrated  without  any  sudden  variation  in  form 
or  optical  properties  of  the  crystals  being  observed;  Georges 
Friedel  *  has  shown  that  analcime  had  not,  at  a  given  tempera- 
ture, an  invariable  dissociation  tension;  let  us  suppose  the  tem- 
perature constant;  in  a  first  equilibrium  state  the  tension  of  the 
water  vapor  which  exists  in  equilibrium  above  the  crystals  has 
the  value  P;  remove  a  portion  of  this  water  vapor;  the  analcime 
will  undergo  a  certain  dehydration  and  the  tension  of  the  water 
vapor  will  increase,  but  only  to  a  value  P',  less  than  P;  and 
so  on;  analcime  is  therefore  not  a  definite  hydrate,  but  only  a 
solid  solution  in  which  water  is  mixed  with  an  anhydrous  silicate. 

Tammann  2  has  extended  Friedel's  observation  to  a  great 
number  of  hydrated  silicates  studied  by  mineralogists,  and  also 
to  hydrated  platino cyanide  of  magnesium: 

MgPt(CN)4+Aq. 

1  G.  FRIEDEL,  Bulletin  de  la  Societe  de  Mineralogie,  v.  19,  p.  363,  1896; 
V.  21,  p.  5,  1898. 

'TAMMANN,  Wiedemann's  Annalen,  v.  73,  p.  16,  1897;  Zeit.  phys.  Chem., 
v.  27,  p.  323,  1898. 


158  THERMODYNAMICS  AND  CHEMISTRY. 

135.  The  existence  of  a  dissociation  tension  does  not  always 
prove  the  existence  of  a  definite  compound.  Dissociation  of 
palladium  hydride. — The  absence  of  a  fixed  dissociation  allows 
also,  in  certain  cases,  to  demonstrate  that  a  body  is  not  a  definite 
compound;  prudence  must  be  exercised  in  concluding  from  the 
existence  of  a  fixed  dissociation  tension  the  existence  of  a  definite 
compound;  the  study  of  the  absorption  of  hydrogen  by  palladium 
will  show  us  that  such  a  conclusion  must  sometimes  be  accepted 
with  caution. 

Following  Troost  and  Hautefeuille,1  let  us  take  palladium, 
kept  at  a  constant  temperature,  and  introduce  hydrogen  into  the 
enclosure  about  this  metal;  the  hydrogen  is  in  part  absorbed  and 
the  tension  attains  a  certain  fixed  value  P]  introduce  more  hydro- 
gen, some  of  this  is  also  absorbed  and  the  pressure  reassumes  the 
value  P;  this  may  go  on  until  the  palladium  has  absorbed  a 
quantity  of  hydrogen  proportional  to  its  mass;  beyond  this,  each 
time  that  hydrogen  is  introduced  it  is  found  that,  after  equilib- 
rium is  established,  the  tension  of  the  gas  attains  a  value  higher 
than  in  the  preceding  operations. 

Troost  and  Hautefeuille  interpreted  these  observations  by  ad- 
mitting that  there  is  first  formed  a  hydride  of  palladium,  of  defi- 
nite composition,  to  which  they  assign  the  formula  Pd2H;  it  is 
only  when  all  the  palladium  would  have  passed  over  to  the  hy- 
dride that  the  hydride  in  its  turn  would  absorb  hydrogen,  forming 
a  solid  hydrogen  solution;  to  the  first  form  of  reaction  corre- 
sponded a  fixed  dissociation  tension,  while  the  second  would  be 
characterized  by  the  absence  of  such  a  tension. 

There  may  be  some  doubt  as  to  the  validity  of  this  conclusion. 

From  the  existence  of  a  fixed  dissociation  tension,  the  only 
consequence  which  we  may  deduce  with  certainty  is  that  the  sys- 
tem is  mono  variant;  and  as  the  system  is  certainly  formed  of  two 
independent  components,  palladium  and  hydrogen,  this  con- 
clusion is  equivalent  to  the  following:  the  system  is  divided  into 
three  phases. 

To  suppose,  with  Troost  and  Hautefeuille,  that  the  system  is 
composed  of  gaseous  hydrogen,  solid  palladium,  and  hydride  of 

1  TROOST  and  HAUTEFEUILLE,  Ann.  de  Chimie  et  de  Physique,  5th  Series, 
v.  2,  p.  279,  1874. 


MONOVARIANT  SYSTEMS. 


159 


solid  palladium,  is  to  make  an  hypothesis  which  accords  with  this 
certain  consequence,  but  which  is  not  equivalent  to  it;  other 
hypotheses,  in  fact,  accord  equally  well  with  this  consequence; 
such,  for  example,  as  the  following  made  by  Roozboom  and  Hoit- 
sema:1  the  solid  mass  that  the  system  includes  would  be  formed 
by  the  juxtaposition  of  two  solid  solutions,  containing  each  a 
different  quantity  of  hydrogen,  comparable  consequently  to  the 
unequally  concentrated  layers  into  which  a  liquid  mixture  of  water 
and  ether  is  divided. 

May  one  decide  between  these  hypotheses? 

If  we  examine  the  question  more  closely,  we  come,  as  is  easy 
to  show,  to  results  which  differ  according  to  the  hypothesis  made 
and  which  may  be  compared  with  experimental  data. 

Let  us  take,  at  a  given  temperature  T,  a  given  mass  of  palla- 
dium, a  gramme  for  example;  also  take  two  rectangular  coor- 


Fio.  37. 


*'  •*      S'  S'S 
PTG.  38. 


7/1 


dinate  axes  (Figs.  37  and  38),  and  along  the  axis  of  abscissae  lay 
off  the  mass  m  of  hydrogen  absorbed  by  a  gramme  of  palladium, 
while  along  the  axis  of  ordinates  lay  off  the  pressure  n  of  the 
hydrogen  in  the  enclosure. 

Let  us  first  trace  the  consequences  of  the  hypothesis  of  Troost 
and  Hautefeuille. 

As  long  as  the  pressure  ?  is  less  than  the  dissociation  tension 
P  of  the  hydrogenized  palladium  at  the  temperature  T,  the  palla- 
dium does  not  absorb  hydrogen;  the  representative  point  de- 
scribes a  segment  OP  of  the  line  On. 

1  HOITSEMA,  Archives  neerlandaises  d.  sciences  exactes  et  naturelles,  v.  30, 
p.  44,  1895;  Zeit.  f.  phys.  Chem,,  v.  17,  p.  1,  1895. 


160  THERMODYNAMICS  AND  CHEMISTRY. 

From  the  moment  the  pressure  exceeds  the  value  P,  the 
hydride  of  palladium  commences  to  be  formed;  as  an  increasingly 
greater  fraction  of  our  mass  of  palladium  passes  over  to  the 
state  of  hydride,  just  so  fast  the  mass  m  of  hydrogen  absorbed 
increases,  while  the  tension  of  hydrogen,  once  equilibrium  is 
established,  remains  constant  and  equal  to  P;  the  representative 
point  describes  a  segment  of  the  straight  line  PA,  parallel  to  OM . 

This  goes  on  until  the  instant  that  the  gramme  of  palladium 
employed  has  passed  entirely  over  to  the  state  of  hydride;  at 
this  moment  the  mass  m  of  hydrogen  absorbed  has  a  definite 
value  ft,  which  is  the  mass  that  must  be  combined  with  a  gramme 
of  palladium  to  obtain  the  substance  Pd2H,  about  2|^  gramme. 

From  this  moment  the  palladium  hydride  as  a  solid  solution 
absorbs  hydrogen;  the  pressure  of  the  hydrogen  in  the  system 
in  equilibrium  does  not  guard  a  fixed  value;  it  increases  at  the 
same  time  as  the  mass  of  hydrogen  absorbed  by  the  palladium 
hydride;  the  representative  point  describes  a  straight  line  AB 
which  rises  from  left  to  right.  ,  ,* 

If  we  repeat  the  same  operations  at  other  temperatures,  T't 
T",  more  and  more  elevated,  we  shall  obtain  the  lines  OP'A'B', 
OP"A"B",  analogous  to  OPAB]  the  lines  PA,  P'A',  P"A", 
parallel  to  the  axis  Om,  will  be  higher  and  higher,  because  to  the 
more  and  more  elevated  temperatures,  T,  Tf,  T" ,  correspond  the 
increasing  dissociation  tensions  P,  P' ,  P"  of  palladium  hydride; 
but  the  points  A,  A' ,  A"  lie  all  on  the  same  parallel  to  the  axis 
Or,  for  their  abscissa  has  a  value  //  which  in  nowise  depends  upon 
the  temperature. 

Let  us  now  reconsider  the  same  operations  and  represent  the 
results  according  to  the  hypothesis  of  Roozboom  and  Hoitsema 
(Fig.  38). 

At  a  given  temperature  T  the  hydrogenized  palladium  may. 
exist  in  two  solid  solutions  of  different  composition;  when  the 
hydrogenized  palladium  thus  separated  exists  in  the  presence 
of  an  atmosphere  of  hydrogen,  one  has  to  do  with  a  monovariant 
system;  where  such  a  system  is  in  equilibrium  at  a  temperature 
T,  not  only  has  the  tension  of  the  hydrogen  a  value  P  which  de- 
pends on  the  single  temperature  T,  but  also  the  two  solid  solutions 
have  compositions  entirely  fixed  by  the  knowledge  of  the  one 


MONOVARIANT  SYSTEMS.  161 

temperature  T;  in  the  first;  containing  the  less  hydrogen,  a  gramme 
of  palladium  is  united  with  s  grammes  of  hydrogen;  in  the  second, 
a  gramme  of  palladium  unites  with  S  grammes  of  hydrogen;  con- 
stant for  a  given  temperature,  the  numbers  s  and  S  vary  with  the 
temperature. 

At  a  given  temperature  T  let  us  take  a  gramme  of  palladium 
and  increase  the  mass  m  of  hydrogen  absorbed  by  this  palladium. 

While  m  is  less  than  s,  the  hydrogenized  palladium  is  com- 
posed of  a  single  solid  solution;  the  system,  which  includes  only 
two  phases,  is  bi variant;  it  does  not  have  a  fixed  dissociation 
tension;  the  tension  of  the  gaseous  hydrogen  for  the  system  in 
equilibrium  increases  with  the  amount  of  hydrogen  held  in  the 
solid  solutions;  the  representative  point  describes  a  curve  Oa 
which  rises  from  left  to  right. 

When  the  mass  m  of  hydrogen  absorbed  by  a  gramme  of  palla- 
dium attains,  then  exceeds  s,  the  hydrogenized  palladium  divides 
into  two  solid  solutions,  the  one  of  s,  the  other  of  S  grammes  of 
hydrogen;  when  the  mass  m  increases,  the  two  quantities  s  and 
S  remain  constant,  but  the  mass  of  the  first  solution  decreases 
and  the  second  increases;  the  system  is  mono  variant;  the  ten- 
sion of  hydrogen  keeps  a  constant  value  P  and  the  representative 
point  describes  a  segment  a  A  of  a  straight  line  parallel  to  Om; 
the  two  extremities  a  and  A  of  this  segment  have  for  abscissae  s 
and  S  respectively. 

When  the  mass  m  of  hydrogen  absorbed  by  a  gramme  of  palla- 
dium attains,  then  exceeds  S,  the  hydrogenized  palladium  forms 
but  a  single  solid  solution;  the  system  again  becomes  bivariant; 
the  tension  of  the  hydrogen  increases  with  the  richness  of  the  solid 
solution;  the  representative  point  describes  a  curve  AB  which 
rises  from  left  to  right. 

Let  us  repeat  the  same  operations  at  the  temperatures  T,  T't 
T",  higher  and  higher;  we  shall  obtain  a  series  of  analogous  lines 
OaAB,  Oa'A'B',  Oa"A"B",  about  which  we  may  make  the  follow- 
ing remarks: 

The  segments  aA,  a' A',  a" A",  parallel  to  the  axis  Om,  are 
more  and  more  elevated,  because  to  the  increasingly  higher  tem- 
peratures T,  T',  T"  correspond  greater  and  greater  transforma- 
tion tensions  P,  P',  P". 


162  THERMODYNAMICS  AND  CHEMISTRY. 

The  origins  of  these  segments  a,  a',  a"  have  abscissae  s,  s',  s" 
different  from  zero  and  differing  among  themselves. 

The  extremities  A,  Af,  A"  of  these  segments  have  abscissae 
S,  S',  S"  which  differ  among  each  other. 

It  suffices  now  to  compare  Figs.  37  and  38  to  see  that  experi- 
ment will  allow  deciding  between  the  hypothesis  of  Troost  and 
Hautefeuille  and  that  of  Roozboom  and  Hoitsema.  The  experi- 
ment has  been  tried  by  the  latter;  the  curves  that  they  have  ob- 
tained, absolutely  irreconcilable  with  the  disposition  of  Fig.  37, 
possess,  on  the  contrary,  differences  from  Fig.  38  which  are  easily 
explicable. 

136.  Robin's  law. — In  order  that  a  mono  variant  system  taken 
at  a  certain  temperature  and  under  a  certain  pressure  may  be 
in  equilibrium,  it  is  necessary  that  this  pressure  be  equal  to  the 
transformation  tension  relative  to  this  temperature;  in  other 
words,  it  is  necessary  that  the  representative  point  which  has 
for  abscissa  this  temperature  and  for  ordinate  this  pressure  b<> 
located  on  the  curve  of  transformation  tensions. 

What  happens  if  the  representative  point  having  tempera- 
ture and  pressure  as  ordinate  is  not  on  the  curve  of  transforma- 
tion tensions?  A  very  simple  rule  answers  this  question;  it  is  due 
to  G.  Robin,1  and  is  thus  stated: 

//  the  representative  point  is  above  the  curve  of  transformation 
tensions,  every  transformation  taking  place  at  the  temperature  and 
under  the  pressure  considered  is  accompanied  by  a  diminution  in 
volume  of  the  system;  the  opposite  takes  place  if  the  representative 
point  is  below  the  curve  of  transformation  tensions. 

Take,  for  example,  a  m  novariant  system  formed  by  a  liquid 
in  contact  with  its  vapor;  at  the  temperature  T  the  tension  of 
saturated  vapor  has  the  value  P;  if  the  pressure  n  has  a  value 
greater  than  P,  the  temperature  remaining  at  T,  the  system  will 
be  the  seat  of  a  modification  accompanied  by  a  lessening  in  volume, 
that  is,  by  a  condensation  of  vapor;  if,  on  the  contrary,  at  the 
same  temperature  T,  the  pressure  is  less  than  the  tension  P  of 
saturated  vapor,  the  system  will  be  the  seat  of  a  modification 


1  G.  ROBIN,  Bulletin  de  la  Societe  philomathique,  7th  Series,  v.  4,  p.  24, 
1879. 


MONOVARIANT  SYSTEMS.  163 

accompanied  by  an  increase  in  volume,  that  is  to  say,  of  vapori- 
zation of  the  liquid. 

Let  us  take  as  a  second  example  a  monovariant  system  formed 
of  calcium  carbonate,  carbonic  acid  gas,  and  lime;  denote  by  P 
the  dissociation  tension  of  calcium  carbonate  for  the  temperature 
T;  if,  at  the  same  temperature,  the  pressure  is  greater  than  Pf 
the  system  will  be  the  seat  of  a  reaction  accompanied  by  a  de- 
crease in  volume,  that  is,  of  a  combination  of  carbonic  acid  gas 
with  lime;  if,  on  the  contrary,  the  pressure  is  less  than  P,  the 
system  will  be  the  seat  of  a  reaction  accompanied  by  an  increase 
in  volume,  or  of  dissociation  of  calcium  carbonate. 

The  accordance  of  these  conclusions  with  the  facts  is  beyond 
criticism. 

137.  Moutier's  Law. — Some  years  before  Robin  stated  the  rule 
which  permits  predicting  the  nature  of  the  modification  which  is 
produced  in  a  monovariant  system  when  the  representative  point 
is  above  or  below  the  curve  of  transformation  tensions,  J.  Moutier 
had  stated  an  analogous  rule;  the  latter  predicts  the  nature  of 
the  modification  of  which  the  system  is  the  seat  according  as  the 
representative  point  is  to  the  left  or  to  the  right  of  the  curve  of 
transformation  tensions. 

This  rule  is  as  follows : 

Let  K  be  a  pressure  arbitrarily  given  and  6  the  transformation 
point  at  this  pressure.  If,  at  the  pressure  TT,  the  temperature  T  has 
a  value  less  than  6,  the  representative  point  is  to  the  left  of  the  curve 
of  transformation  tensions;  in  these  conditions  the  system  is  the 
seat  of  a  certain  modification;  taking  place  at  the  same  pressure  x 
and  at  the  temperature  d,  this  modification  will  liberate  heat. 

If,  at  the  pressure  n,  the  temperature  T  has  a  value  greater  than  6, 
the  representative  point  is  to  the  right  of  the  curve  of  transformation 
tensions;  in  these  conditions  the  system  is  the  seat  of  a  modification; 
accomplished  at  the  same  pressure  n  and  temperature  6,  this  modi- 
fication would  absorb  heat. 

For  example,  at  a  given  pressure  and  at  the  boiling-point 
corresponding  to  this  pressure,  vaporization  of  the  liquid  absorbs 
heat;  the  condensation  of  the  vapor  liberates  heat;  therefore, 
at  this  same  pressure,  for  temperatures  less  than  the  boiling-point 


164  THERMODYNAMICS  AND  CHEMISTRY. 

the  vapor  condenses,  while  at  temperatures  higher  than  the  boiling- 
point  the  liquid  vaporizes. 

At  a  given  pressure  and  at  the  fusing-point  corresponding 
to  this  pressure,  the  fusion  of  a  solid  absorbs  heat,  while  the  freez- 
ing of  the  liquid  sets  heat  free;  therefore,  at  this  same  pressure, 
for  temperatures  less  than  the  fusing-point  the  liquid  freezes, 
while  at  temperatures  higher  than  the  fusing-point  the  Bolid 
melts. 

138.  False  equilibria  in  monovariant  systems.— These  exam- 
ples show  how  easy  it  is,  in  most  cases,  to  apply  Robin's  or  Mou tier's 
rule  to  a  given  monovariant  system;  nevertheless,  when  pre- 
dictions of  the  modifications  of  a  given  monovariant  system  are 
sought  by  aid  of  these  rules,  one  must  not  lose  sight  of  what  has 
been  said  at  the  end  of  Chapter  VI  (Arts.  98  and  99)  concerning 
the  phenomena  of  false  equilibrium. 

When  thermodynamics  indicates  that  a  modification  is  im- 
possible, such  a  modification  cannot  take  place;  but  when  it 
announces  that  a  change  should  occur,  it  may  happen  that  no 
change  is  produced,  and  the  system  remains  in  equilibrium. 

Striking  examples  of  this  general  remark  may  be  found. 

Moutier's  rule  teaches  that  at  a  temperature  less  than  the 
fusing-point  the  solid  cannot  melt,  while  the  liquid  should  freeze; 
and  in  fact  at  a  temperature  below  the  fusing-point  the  solid 
never  does  pass  into  the  liquid  state;  but  it  may  very  well  happen 
that  the  liquid  does  not  freeze  and  yet  remains  in  equilibrium; 
it  is  in  general  sufficient,  in  order  to  observe  this  phenomenon 
of  sur fusion,  to  avoid  shocks  and  especially  the  introduction 
of  a  particle  of  the  so  id  which  the  liquid  would  give  in  freezing. 

Robin's  rule  teaches  that  under  a  pressure  higher  than  the 
tension  of  saturated  vapor  corresponding  to  the  temperature  of 
the  experiment,  the  vapor  should  pass  into  the  liquid  state;  as 
a  fact,  if  compression  is  carefully  brought  about  and  if  the  vapor 
is  free  from  all  dust  and  liquid  drops,  it  is  easy  to  obtain  a  vapor 
under  a  higher  pressure  than  the  tension  of  saturated  vapor  with- 
out condensation;  this  was  first  observed  by  Coulier  and  after- 
wards studied  by  Wiillner  and  Grotian. 

It  suffices  for  the  moment  to  mention  these  phenomena,  which 
will  be  studied  farther  on  (see  Chap.  XVII,  Arts.  274-278). 


MONOVARIANT  SYSTEMS.  165 

139.  Another  form  of  Moutier's  law. — As  simple  as  Robin's 
rule,  and  also  antedating  it,  Moutier's  rule  surpasses  the  former 
by  the  originality  of  views  that  it  introduces  into  chemical  statics. 

Under  a  pressure  TT  let  0  be  the  value  of  the  transformation 
point  of  a  mono  variant  system;  let  us  consider  a  reaction  taking 
pk.ce  in  this  system  under  the  same  pressure  n  and  at  the  tem- 
perature T,  different  from  0:  this  reaction  calls  into  play  a  cer- 
tain quantity  of  heat,  which  depends  upon  the  temperature  T, 
as  we  have  seen  (Chap.  Ill,  Art.  41);  if,  therefore,  without  chang- 
ing the  pressure,  we  « ause  the  temperature  T  to  vary,  making  it 
approach  6,  the  value  of  the  quantity  of  heat  set  free  by  the  re- 
action will  vary,  and  even  its  sign  may  change;  this  last  circum- 
stance will  certainly  not  be  produced  if  the  temperatures  T  are  not 
too  far  distant  from  the  transformation  point  6',  let  us  suppose 
that  this  is  not  produced  for  the  monovariant  systems  which  we 
shall  study  and  in  the  conditions  under  which  we  study  them; 
Moutier's  rule  may  then  be  stated : 

At  a  given  pressure,  every  change  produced  in  a  monovariant 
system  at  a  temperature  less  than  the  transformation  point  is  accom- 
panied by  a  liberation  of  heat;  every  modification  produced  at  a 
temperature  higher  than  the  transformation  point  is  accompanied 
by  an  absorption  of  heat. 

140.  Corollary  to  this  law. — Moutier  deduced  a  consequence 
which  is  almost  identical  with  this  proposition,  but  which  has  the 
advantage  of  better  emphasizing  its  importance: 

Imagine  that  in  the  same  monovariant  system,  under  the  same 
pressure,  but  at  two  different  temperatures,  two  reactions,  the  inverse 
of  each  other,  are  observed;  the  reaction  taking  place,  at  the  lower 
temperature  is  exothermic,  the  one  produced  at  the  higher  tempera- 
ture is  endothermic. 

Thus,  in  the  same  monovariant  system  formed  of  cupric  oxide, 
cuprous  oxide,  and  oxygen,  and  under  the  same  pressure,  two 
inverse  reactions  may  be  observed;  at  a  certain  temperature, 
oxidation  of  curpous  oxide;  at  a  higher  temperature,  the  disso- 
ciation of  cupric  oxide;  the  dissociation  of  cupric  oxide  absorbs 
heat,  the  oxidation  of  cuprous  oxide  liberates  heat. 

141.  Consequence  relative  to  very  low  temperatures;  the  prin- 
ciple of  maximum  work  is  exact  at  these  temperatures. — Let 


166  THERMODYNAMICS  AND  CHEMISTRY. 

us  suppose  for  an  instant  that  we  have  never  to  consider  other 
than  monovariant  systems,  and  let  us  see  what  follows  from  the 
preceding  proposition. 

If;  under  a  given  pressure,  as  that  of  the  atmosphere,  the 
temperature  was  sufficiently  lowered  so  that  it  became  less  than 
all  the  transformation  points  of  the  systems  considered,  it  would 
be  impossible  to  observe  reactions  other  than  those  accompanied 
by  a  liberation  of  heat;  all  the  spontaneous  decompositions  would 
be  decompositions  of  endothermic  compounds,  all  the  sponta- 
neous syntheses  would  be  syntheses  of  exothermic  compounds. 

On  the  contrary,  if  the  temperature  were  raised  sufficiently  to 
become  higher  than  all  the  transformation  points  of  the  systems 
studied,  all  the  possible  reactions  would  be  accompanied  by  an 
absorption  of  heat;  all  spontaneous  decompositions  would  be 
decompositions  of  exothermic  compounds,  all  spontaneous  syn- 
theses would  be  syntheses  of  endothermic  compounds. 

In  other  terms',  at  a  sufficiently  low  temperature  the  principle 
of  maximum  work  would  rigorously  apply, 

142.  Consequence  for  high  temperatures. — On  the  other  hand, 
at  sufficiently  high  temperatures  this  principle  would  be  turned 
upside  down  and  replaced  by  its  opposite;  it  is  therefore  clear 
that,  in  creating  the  chemistry  of  high  temperatures,  H.  Sainte- 
Claire  Deville  had  to  meet  a  host  of  facts  irreconcilable  with  the 
principle  set  forth  by  J.  Thomsen;  we  see  why  he  could  obtain 
the  dissociation  of  a  great  number  of  exothermic  compounds, 
such  as  water  or  carbonic  acid  gas  (Arts.  49,  50,  51) ;  it  is  as  easily 
understood  how  his  disciples  could,  at  very  high  temperatures, 
reproduce  endothermic  substances  which  are  spontaneously  de- 
stroyed at  much  lower  temperatures,  as  we  shall  see  in  the  follow- 
ing chapter. 

Evidently  these  conclusions  are  not  logically  established  by 
what  precedes  except  on  the  condition  to  suppose  monovariant 
all  the  chemical  systems  found  in  nature;  but  their  generality, 
which  will  be  demonstrated  in  the  following  chapter,  may  already 
be  anticipated;  as  long  ago  as  1877,  Moutier  affirmed  it  with 
great  clearness.1 

1  J.  MOUTIER,  Bulletin  de  la  Socictg  philomathique,  3d  Series,  v.  i,  p.  96, 
1877.  This  important  contribution  to  the  history  of  chemical  mechanics 


MONOVARIANT  SYSTEMS. 


167 


These  results  transformed  profoundly  the  ideas  previously  held 
concerning  the  opposition  which  exists  between  exothermic  and 
endo  thermic  reactions. 

For  the  chemists  of  the  beginning  of  the  nineteenth  century 
every  combination  was  exothermic,  every  decomposition  endo- 
thermic. 

For  the  thermo chemists  who  accepted  without  restriction  the 
principle  of  maximum  work  an  exothermic  reaction  was  one 
susceptible  of  producing  itself;  an  endo  thermic  reaction  could 
not  take  place  without  the  aid  of  external  energy. 

In  modern  chemical  mechanics  an  exothermic  reaction  is  one 
susceptible  of  producing  itself  at  low  temperature;  an  endo- 
thermic  reaction  is  one  that  may  produce  itself  at  high  tempera- 
ture. 

143.  Similarity  of  Moutier's  and  Robin's  laws.  Form  of  the 
curve  of  transformation  tensions. — Comparing  the  two  laws  stated 
by  J.  Moutier  and  by  G.  Re  bin,  we  are  immediately  led  to  im- 
portant conclusi  ns. 

Consider  a  mono  variant  system;    let  M  (Fig.  39)  be  a  point 
of  the  transformation  curve  of  this  system;     _ 
6,    P   are   abscissa    and   ordinate    of    this 
point. 

Let  us  suppose  that  the  system  studied  n 
can  present  only  two  kinds  of  modifications, 
the  inverse  of  each  other;  for  example,  the 
fusion  of  a  solid  and  the  freezing  of  a  liquid, 
or  the  combination  of  carbonic  acid  gas  with 
lime  and  the  dissociati  n  of  calcium  car- 
bonate. 

Among  the  modifications  which  this  system  may  have,  there 
are  some  which,  taking  place  at  the  temperature  6  and  under 
the  pressure  P,  would  absorb  heat;  there  are  others,  the  inverse 
of  the  former,  which  liberate  heat.  Let  us  choose  one  of  these 
latter. 

Taking  place  at  the  temperature  6  and  pressure  P,  it  would 
determine  a  certain  variation  of  the  volume  of  the  system;  this 

is  reproduced  in  Duhem's  Introduction  a  la  Mecanique  Chimie,  p.  147  (Gand, 
1893). 


T    0 
FIG.  39. 


168  THERMODYNAMICS  AND  CHEMISTRY. 

variation  may  be  a  decrease,  and  we  shall  call  this  the  first  case;  it 
may  be  an  increase,  and  we  shall  call  this  the  second  case. 

When  the  temperature  and  pressure  change  gradually,  the 
quantity  of  heat  developed  by  a  definite  change  varies  in  a  con- 
tinuous manner,  and  this  is  also  true  for  the  change  in  volume 
accompanying  this  modification;  one  may  therefore  always  sup- 
pose this  change  of  temperature  and  pressure  small  enough  so 
that  it  does  not  necessitate  any  change  of  sign  either  in  the  quan- 
tity of  heat  involved  in  the  modification  or  in  the  variations  of 
the  volume  which  it  causes. 

Applied  to  the  modification  which  we  have  chosen,  this  re- 
mark may  be  put  into  the  following  form: 

About  the  point  M  it  is  always  possible  to  draw  a  region  D 
small  enough  so  that  it  has  the  following  property:  at  the  tem- 
perature T  and  pressure  II  which  serve  as  coordinates  for  any 
point  /*,  in  this  region,  the  modification  considered  would  cause  a 
liberation  of  heat;  there  will  be  a  diminution  of  volume  for  the 
first  case,  and  an  increase  of  volume  for  the  second  case. 

From  now  on,  let  us  consider  only  the  points  of  this  region 
and  apply  Mou tier's  rule  to  them;  first,  this  shows  us  that  at  the 
temperature  T  and  pressure  II  which  serve  as  coordinates  to  one 
of  these  points  /*,  the  modification  considered  cannot  be  pro- 
duced, unless  the  point  /x  is  to  the  left  of  the  curve  of  transforma- 
tion tensions. 

Let  us  now  make  use  of  Robin's  rule;  for  the  first  case  the 
modification  may  be  produced  at  the  temperature  T  and  pressure 
II  only  if  the  point  fi  is  above  the  curve  of  transformation  tensions ; 
in  the  second  case,  on  the  contrary,  it  can  be  produced  only  if  the 
point  «  is  below  the  curve  of  transformation  tensions. 

In  order  that  these  conclusions  may  be  in  accord,  it  is  neces- 
sary that  the  portion  of  the  region  D  which  is  to  the  left  of  the  curve 
of  transformation  tensions  be,  at  the  same  time,  in  the  first  case 
above  this  curve,  and  in  the  second  case  below,  which  leads  to  the 
theorem: 

Let  6  and  P  be  the  coordinates  of  a  point  M  on  the  curve  of  trans- 
formation tensions  of  a  monovariant  system;  choose  a  modifica- 
tion of  this  system  which  liberates  heat  when  it  is  supposed  to  take 
place  at  the  temperature  6  and  the  pressure  P;  if  this  modification 


MONOVARIANT  SYSTEMS. 


169 


o         e 

FIG.  40. 


T      O 


e 

FIG.  41. 


is  accompanied  by  a  decrease  in  volume  of  the  system,  the  curve  of 
transformation  tensions  rises  from  left  to  right  in  the  neighborhood 
of  the  point  M  (Fig.  40);  if  this  modification  is  accompanied  by 
an  increase  in  volume  of  the  rr 

C*  /     TT 

system,  the  curve  of  transforma- 
tion tensions  descends  from  left 
to  right  near  the  point  M  (Fig.  41). 
Let  us  apply  this  theorem  to 
some  simple  cases. 

At  a  given  temperature  and 
under  the  tension  of  vapor  satu- 
rated at  this  temperature,  the  condensation  of  vapor  to  the  liquid 
state  is  accompanied  by  a  liberation  of  heat  and  a  decrease 
in  volume;  the  curve  of  tensions  of  saturated  vapor  should  rise 
from  left  to  right;  the  tension  of  saturated  vapor  is  the  greater 
the  higher  the  temperature. 

At  a  given  temperature  and  under  a  pressure  equal  to  the 
dissociation  tension  for  this  temperature,  the  combination  of 
carbonic  acid  gas  with  lime  is  accompanied  by  a  decrease  in  volume 
of  the  system  and  by  a  liberation  of  heat;  the  curve  of  dissocia- 
tion tensions  of  calcium  carbonate  should  rise  from  left  to  right. 

Under  a  given  pressure  and  at  a  temperature  equal  to  the 
fusing-point  for  this  pressure,  the  freezing  of  a  liquid  sets  heat  free. 

Most  liquids  decrease  in  volume  on  freezing;  the  curve  of 
transformation  tensions  should  therefore  rise  from  left  to  right: 
some  liquids,  such  as  water,  increase  in  volume  in  freezing;  for 
these  substances  the  curve  of  transformation  tensions  should 
descend  from  left  to  right;  whence  the  following  proposition, 
which  has  been  verified  by  a  great  number  of  observers: 

For  almost  all  solids  whose  fusion  is  accompanied  by  expansion, 
the  fusing-point  rises  at  the  same  time  as  the  pressure  supported  by  the 
system  increases;  for  solids,  such  as  ice,  whose  fusion  is  accompanied 
by  contraction,  the  fusing-point  is  lowered  when  the  pressure  increases. 

We  shall  go  farther  and  find  the  value  of  the  tangent  to  the 
curve  of  transformation  tensions;  but  in  order  to  be  able  to  state 
the  important  equation  which  determines  the  value  of  this  quan- 
tity, we  must  first  speak  of  the  modifications  that  a  monovariant 
system  may  be  supposed  to  assume. 


170  THERMODYNAMICS  AND  CHEMISTRY. 

144.  In  every  monovariant  system  there  may  be  two  modi- 
fications, the  inverse  of  each  other,  which  change  the  masses 
of  the  phases  without  changing  their  composition. — In  certain 
monovariant  systems  only  two  modifications  can  be  observed, 
the  inverse  of  each  other;  thus,  in  a  system  which  encloses  a  liquid 
and  its  vapor,  only  the  vaporization  of  the  liquid  or  the  condensa- 
tion of  the  vapor;  in  a  system  containing  a  solid  and  the  liquid 
resulting  from  its  fusion,  only  the  fusion  of  the  solid  or  the  freezing 
of  the  liquid  can  be  observed;  in  a  system  containing  carbonate 
of  calcium,  lime,  and  carbonic  acid  gas,  one  may  observe  only 
the  combination  of  carbonic  acid  with  lime  or  the  dissociation  of 
the  carbonate  of  calcium. 

In  the  various  systems  we  have  mentioned  each  phase  is  a 
substance  of  definite  composition;  it  is  therefore  evident  that 
the  two  opposite  kinds  of  modification,  which  may  be  produced, 
change  the  masses  of  the  various  phases  without  changing  their 
composition. 

Some  other  monovariant  systems  are  more  complicated. 

Take,  for  example,  a  system  whose  independent  components  are 
water  and  sodium  chloride  and  which  is  divided  into  three  phases : 
solid  chloride  of  sodium,  an  aqueous  solution,  and  water  vapor; 
it  may  be  supposed  that  sodium  chloride  dissolves  without  con- 
densation of  water  vapor,  that  water  vapor  condenses  without 
dissolving  sodium  chloride,  that  when  a  mass  of  sodium  chloride 
is  dissolved  and  a  mass  of  water  vapor  condensed,  the  ratio  of 
these  two  masses  may  have  any  value  whatever;  it  may  similarly 
be  imagined  that  the  system  undergoes  any  modifications  the 
opposite  of  the  preceding. 

It  is  quite  clear  that  in  general  these  modifications  change 
the  composition  of  the  solution  of  sodium  chloride;  if;  for  example, 
sodium  chloride  is  dissolved  without  condensing  water  vapor,  the 
solution  becomes  more  concentrated;  if  one  condenses  water  vapor 
without  dissolving  sodium  chloride,  the  solution  becomes  more 
dilute. 

It  is  always  possible  to  imagine  two  kinds  of  modification,  the 
opposite  of  each  other,  which  do  not  change  the  concentration  of 
the  sodium  chloride  solution ;  the  first  of  these  two  kinds  of  modi- 
fication consists  in  dissolving  a  certain  mass  of  sodium  chloride 


MONOVARIANT  SYSTEMS.  171 

and,  at  the  same  time,  in  condensing  a  certain  mass  of  water,  the 
ratio  of  the  first  mass  to  the  second  being  exactly  equal  to  the 
concentration  of  the  solution;  the  second  modification  is  to  take 
from  the  solution  a  mass  of  sodium  chloride  and  a  mass  of  water 
vapor,  the  ratio  of  the  first  mass  to  the  second  being  again  equal 
to  the  concentration  of  the  solution. 

This  remark  may  be  generalized  into  the  following  proposition: 
In  any  monovariant  system,  two  kinds  of  modification  the  oppo- 
site of  each  other  may  be  imagined  which  change  gradually  the  mass 
of  each  of  the  phases  without  altering  the  composition  of  any  of  them. 

145.  The  equilibrium  of  a  monovariant  system  is  indifferent. 
— This  proposition  gives  rise  to  a  consequence  which  should  be 
noted. 

Let  T  be  any  temperature  whatever;  in  order  that  a  mono- 
variant  system  be  in  equilibrium  at  this  temperature  T,  it  is  neces- 
sary and  sufficient  that  the  pressure  have  a  definite  value  P, 
which  is  the  transformation  tension  for  this  temperature,  and 
that  each  of  the  phases  into  which  the  system  is  divided  have  a 
definite  composition;  then  it  is  easy  to  see  that  if  the  temperature 
T  and  the  pressure  P  are  kept  constant,  the  monorariant  system  is 
in  indifferent  equilibrium. 

One  may,  in  fact,  withcut  changing  either  the  temperature 
T  or  the  pressure  P,  impose  on  the  system  a  modification  which 
changes  the  masses  of  the  various  phases  without  changing  the 
composition  of  any  of  them;  such  a  modification  will  not,  there- 
fore, disturb  the  equilibrium  of  the  system. 

Let  us  take,  for  instance,  a  system  which  encloses  a  liquid  and 
its  vapor;  at  a  given  temperature  this  system  is  in  equilibrium 
if  the  pressure  is  equal  to  the  tension  of  saturated  vapor;  without 
changing  temperature  or  pressure,  condense  a  certain  mass  of 
vapor  or  vaporize  a  certain  mass  of  liquid;  the  system  is  modi- 
fied, but  all  the  states  thrpugh  which  it  passes  are  equilibrium 
states:  the  equilibrium  of  a  system  formed  by  a  liquid  and  its 
vapor  is  therefore  a  state  of  indifferent  equilibrium. 

146.  Law  of  Clapeyron  and  Clausius.— The   conclusion  that 
we  have  just  drawn  is,  besides,  as  we  shall  see,  not  the  only  one 
that  may  be  deduced  from  the  existence  of  modifications  which 
leave  unaltered  the  composition  of  each  of  the  phases    of    a 


172  THERMODYNAMICS  AND  CHEMISTRY. 

mono  variant  system;  it  is  from  the  consideration  of  such  modifi- 
cations that  we  are  going  to  determine  the  tangent  to  the  curve 
of  transformation  tensions. 

Take,  in  a  monovariant  system,  a  point  M  (Fig.  42)  on  the 
curve  C  of  transformation  tensions;  let  6  and  P  be  the  coordi- 
nates of  this  point;  suppose,  what  has 
not  been  necessary  to  this  time,  that  6 
is  not  the  temperature  on  any  thermome- 
ter,' but  the  absolute  temperature;  draw 
through  M  a  tangent  Mt  to  the  curve  C; 
this  tangent  makes  with  the  line  OT  or 
with  its  parallel,  Mr,  an  angle  a;  it  is 
the  trigonometrical  tangent  of  this  angle 


e  T  which  we  wish  to  know. 

Fro-  42.  Among  the  modifications  which  change 

the  masses  of  the  phases  into  which  a  monovariant  system  is 
divided  without  changing  the  composition  of  any  of  these  phases, 
there  are  necessarily  some  which  at  the  temperature  6  and  pres- 
sure P  liberate  heat.  Take  one  of  these  modifications  and  let  Q 
be  the  quantity  of  heat,  positive  by  definition,  which  it  liberates. 
Denote  by  V  the  increase  in  volume  of  the  system  due  to  this 
modification;  if  this  change  is  accompanied  by  a  contraction,  V 
is  negative. 

Thermodynamics  gives  us  the  following  equation: 

rr    /") 

(1)  tma=-j-y, 

E  being  the  mechanical  equivalent  of  heat. 

This  formula  agrees,  as  is  easily  seen,  with  the  propositions 
which  we  have  obtained  by  combining  Mou tier's  and  Robin's 
rules. 

If  the  modification  considered  is  accompanied  by  a  decrease  in 
volume,  V  is  negative,  tan  a  is  positive,  and  the  curve  of  trans- 
formation tensions  rises  from  left  to  right. 

If  the  modification  considered  is  accompanied  by  an  increase 
in  volume,  V  is  positive,  tan  a  negative,  and  the  curve  of  trans- 
formation tensions  descends  from  left  to  right. 

This  relation  is  one  of  the  oldest  known  to  thermodynamics; 


MONOVARIANT  SYSTEMS.  173 

it  already  existed,  although  in  an  incomplete  form,  in  the  com- 
mentary given  by  Clapeyron  on  the  work  of  Sadi  Carnot  on  the 
motive  power  of  heat;  it  was  later  completed  and  demonstrated 
by  R.  Clausius  among  his  earliest  researches  in  thermodynamics. 

Clausius  applied  it  in  the  first  place  to  the  best  known  of  mono- 
variant  systems,  that  formed  by  a  liquid  and  its  vapor.  Let  us 
see  what  form  this  relation  takes  for  such  a  system. 

147.  Application  to  vaporization.  —  As  modification  leaving 
invariable  the  compo  ition  of  the  system  and  producing  a  liber- 
ation of  heat,  we  may  take  the  condensation  of  a  gramme  of  vapor; 
at  the  temperature  6,  under  the  pressure  P  which  is  the  tension 
of  saturated  vapor  at  this  temperature,  the  quantity  of  heat  Q 
liberated  by  this  modification  is  exactly  the  heat  of  vaporization 
of  the  liquid  considered,  at  the  temperature  6.  At  the  tempera- 
ture 6  and  pressure  P  let  v  be  the  volume  of  a  gramme  of  vapor 
and  v'  the  volume  of  a  gramme  of  liquid;  v  is  called  the  specific 
volume  of  the  saturated  vapor  at  the  temperature  T,  and  i/  the  spe- 
cific volume  of  the  liquid  at  the  same  temperature.  The  change 
considered  produces  an  increase  in  volume, 

V=v'-v, 

but  vf  is  less  than  v,  so  that  V  is  negative  and  corresponds  to  a 
decrease  in  volume. 

The  formula  (1)  therefore  becomes 

(2)  tan*=.. 


This  relation  has  in  the  theory  of  saturated  vapors  a  very  great 
importance,  which  we  only  mention  here,  without  entering  into 
detailed  explanations  which  interest  especially  the  physicist. 

148.  Application  to  fusion.  Variation  of  fusing-point  with 
pressure.  —  J.  Thomson  was  the  first  to  remark,  in  1849,  that  a 
relation  similar  in  all  respects  to  equation  (2)  should  apply  to  the 
curve  which  represents  the  variations  of  the  fusing-point  of  a 
substance  with  the  pressure.  To  pass  from  the  preceding  to  the 
present  case  it  suffices  to  substitute  the  word  liquid  for  vapor  and 
the  word  solid  for  liquid;  by  this  substitution  L  becomes  the  heat 
of  fusion,  v  the  specific  volume  of  the  liquid  at  the  point  of  fusion 
0  under  the  pressure  P,  i/  the  specific  volume  of  the  solid  under 
the  same  conditions. 


174  THERMODYNAMICS  AND  CHEMISTRY. 

There  is  a  preliminary  remark  to  be  made: 

Whether  it  is  a  question  of  a  heat  of  vaporization  or  a  heat  of 
fusion,  the  order  of  magnitude  of  L  remains  the  same,  at  least 
under  ordinary  conditions;  this  cannot  be  said  of  the  expression 
(v— v')  by  which  L  is  divided;  the  volume  occupied  by  a  gramme 
of  vapor  is  in  general  incomparably  greater  than  the  volume 
occupied  by  a  gramme  of  the  same  substance  in  the  liquid  or 
solid  state ;  it  is  then  evident  that,  in  the  case  of  fusion,  tan  a  will 
have  a  value  very  much  greater  than  that  assumed  by  this  quan- 
tity for  most  cases  of  vaporization. 

If  the  specific  volume  of  the  liquid  exceeds  that  of  the  solid 
(v— 1/>0),  the  curve  of  fusion  rises  from  left  to  right  very  steeply; 
the  fusing-point  is  the  higher  as  the  pressure  on  the  system  in- 
creases; but  in  order  to  attain  an  appreciable  rise  of  the  fusing- 
point,  a  very  considerable  increase  must  be  given  to  the  pressure. 

If  the  specific  volume  of  the  solid  is  greater  than  that  of  the 
liquid  (vf  —  v>0),  the  curve  of  fusion  rises  from  right  to  left  with 
great  rapidity;  the  fusing-point  is  lowered  with  increase  of  pres- 
sure, but  the  lowering  of  the  fusing-point  is  noticeable  only  with 
a  great  increase  of  pressure. 

This  latter  case  is  the  same  as  that  of  the  fusing-point  of  ice. 

Equation  (2)  not  only  allows  us  to  affirm  that  a  great  change 
of  pressure  produces  a  slight  variation  of  the  fusing-point;  it  also 
permits  calculating,  approximately  but  quite  sufficiently,  what 
variation  of  the  fusing-point  is  produced  by  a  given  increase  in 
pressure;  one  may,  in  fact,  within  certain  limits,  replace  the 
curve  of  fusion  by  its  tangent. 

Denote  by  6Q  (Fig.  43)  the  fusing-point  under  the  pressure 
TTO  for  atmospheric  pressure;  by  #t  the  fusing- 
point  at  the  pressure  T^;  let  M0  be  the  point 
whose  coordinates  are  00,  TTO,  and  M1  the  point 
of  coordinates  0lf  7rt;  in  the  right  triangle  MQfj.Mi 
we  have 


___!»«,  tan  a  =  ^^ = **— ** 


Comparing  this  with  equation  (2),  we  get 


MONOVARIANT  SYSTEMS. 


175 


By  means  of  this  equation  James  Thomson  showed  that  for 
an  increase  in  pressure  from  1  to  8.1  atmospheres  the  fusing-point 
of  ice  would  be  lowered  by  ^  degrees  Fahrenheit;  very  precise 
experiments  made  by  William  Thomson  gave  a  lowering  of  T}T 
degrees. 

Almost  all  substances  other  than  ice  increase  in  volume  upon 
melting;  for  these  substances  the  fusing-point  rises  with  increase 
of  pressure;  by  a  very  ingenious  method  Bunsen  verified  quali- 
tatively this  prediction  of  theory  for  spermaceti  and  paraffine; 
various  physicists  have  extended  this  verification  to  a  great  num- 
ber of  substances. 

Others  have  gone  farther  and  sought  a  quantitative  con- 
firmation for  formula  (2) ;  taking  this  formuk  as  basis,  they  have 
calculated  the  increase  that  the  fusing-point  should  have  for  cer- 
tain substances  for  an  increase  of  one  atmosphere  in  pressure, 
and  they  have  compared  the  number  calculated  with  the  number 
observed;  the  following  table  shows  such  a  series  of  comparisons: 


Substance  studied. 

Rise  of  Fusing-point. 

Observer. 

Observed. 

Calculated. 

Acetic  acid 

0°.  02425 
.0294 
.0187 
.0170 

0°.  02421 
.02936 
.0188 
.0170 

De  Visser 

Demerliac 
ii 

ii 

BenziiiB 

Paratoluidine 

Naphtylamine  or.             ... 

We  therefore  have  very  exact  verifications  of  the  relation  of 
Clapeyron  and  Clausius. 

149.  Application  to  the  allotropic  transformation  of  a  solid 
into  another  solid. — Formula  (2)  evidently  applies  almost  without 
modification  to  the  case  in  which  a  substance  may  exist  in  two  dis- 
tinct solid  forms  a  and  6,  resulting  from  an  allotropic  or  isomeric 
transformation;  suppose  that  the  form  b  passes  into  the  form  a 
with  liberation  of  heat;  under  the  pressure  P  there  exists  a  trans- 
formation point  6;  at  temperatures  lower  than  6  the  form  b 
passes  over  to  a;  above  6,  a  changes  to  b.  When,  at  the  tempera- 
ture 6  and  pressure  P,  a  gramme  of  the  form  b  passes  into  the  form 


176  THERMODYNAMICS  AND  CHEMISTRY. 

a,  L  is  the  quantity  of  heat  set  free;  if  va,  vb  are  the  volumes  occu- 
pied by  a  gramme  of  forms  a  and  b  respectively,  we  may  write 

(4)  ten«-f.i. 

The  heat  of  transformation  L  and  the  difference  (v&— v0)  of 
the  specific  volumes  are,  in  general,  quantities  of  the  same  order 
of  magnitude  as  a  heat  of  fusion  and  the  difference  between  the 
specific  volume  of  the  liquid  and  that  of  the  solid  taking  part  in 
this  fusion;  one  might  therefore  develop  here  considerations  analo- 
gous to  those  we  mentioned  for  the  variation  of  the  fusing-point, 
and  conclude  likewise  that  a  considerable  increase  in  pressure 
will  produce  only  a  slight  change  in  the  transformation  point. 

A  rough  verification  of  formula  (4)  has  been  tried  by  Mallard 
and  Le  Chatelier  by  studying  the  allo  tropic  changes  of  silver 
iodide.  An  investigation  of  the  modifications  of  ammonium 
nitrate  has  given  Silvio  Lussana  a  more  precise  verification  of 
formula  (4),  or  rather  of  the  formula 


which  is  analogous  to  formula  (3)  . 

Crystallized  solid  ammonium  nitrate  occurs  in  four  distinct 
allo  tropic  forms  which  we  shall  denote  by  a,  6,  c,  d;  each  of  these 
forms  may  be  derived  from  the  preceding  with  liberation  of  heat. 

Under  atmospheric  pressure  each  of  these  forms  passes  over 
into  the  following  when  the  temperature  reaches  a  certain  trans- 
formation point;  the  three  transformation  points  have  about  the 
following  values; 


0crf  =  273°  +  125°. 

Corresponding  to  these  transformation  points  are  the  follow 
ing  heats  of  transformation: 

Lab=  5.02  cal. 
Lbc=  5.33    " 
Lcd=  11.86    " 


MONOVARIANT  SYSTEMS. 


177 


The  first  change  is  accompanied  by  a  variation  in  volume  (in 
cubic  centimetres  per  gramme) : 


The  second  modification  is  accompanied  by  a  volume  change 
of  opposite  sign: 

vc-vb=  -0.00854. 

Finally,  for  the  third  change,  the  sign  of  vd—  vcis  known,  but 
not  its  magnitude, 


For  any  one  of  these  modifications  let  60  be  the  transforma- 
tion point  under  atmospheric  pressure,  and  6  this  point  under  a 
pressure  TT;  these  transformation  points  may  be  observed  either 
by  heating  or  cooling  the  substance;  from  which  we  have  two 
distinct  values,  both  deduced  from  observation,  for  the  difference 
(0-0o);  we  shall  denote  them  by  (0-00)hot  and  (0-00)coid;  these 
values  are  given  in  the  following  table  together  with  (6—  #0)caic., 
the  approximate  values  of  (6—6Q)  which  are  deduced  from  equa- 
tion (5). 

It  is  evident  that  the  study  of  the  transformations  a  —  b  and 
b—  c  furnishes  remarkable  confirmations  of  formula  (5);  concerning 
the  transformation  c—  d  for  which  a  quantitative  comparison  is 
impossible,  it  agrees  qualitatively  with  equation  (5). 


Transformation. 

Pressure  77  in 
Atmospheres. 

(0-00)hot 

(0-00)cold 

(0-0)calc. 

0-6 

50 

100 
150 
200 
250 

+  1°.60 
3  .14 
4  .32 
6  .02 
7  .31 

+  1°.43 
2  .82 
4  .65 
5  .89 
7  .40 

+  1°.67 
2  .94 
4  .41 
5  .88 
7  .35 

6-c      , 

50 
100 
150 
200 
25C 

-0°.70 
1   .47 
2  .12 
2  .82 
3  .56 

-0°.70 
1  .40 
2  .10 
2  .80 
3  .55 

c-d 

50 
100 
150 
200 
250 

+  0°.58 
1  .20 
1  .88 
2  .31' 
3  .15 

+  0°.65 
1  .15 
1  .74 
2  .36 
3  .00 

>0 

178  THERMODYNAMICS  AND  CHEMISTRY. 

150.  Application  to  dissociation. — Equation  (1)  is  very  easily 
applied  to  the  case  of  dissociation  of  which  the  dissociation  of 
calcium  carbonate  is  the  type. 

Consider  w  grammes  of  the  compound,  calcium  carbonate  in 
this  case,  equal  to  its  molecular  weight;  in  order  to  form  w 
grammes  of  the  compound  it  is  necessary  to  combine  p  grammes 
of  the  gaseous  component,  here  carbonic  acid,  with  p'  grammes 
of  the  solid  component,  lime. 

Let  us  suppose  the  compound  exothermic;  taking  place  at 
the  temperature  6,  under  a  pressure  equal  to  the  dissociation  ten- 
sion P,  the  formation  of  a  gramme  of  the  compound  sets  free  L 
calories;  the  formation  of  w  grammes  of  the  compound  liberates 
Q  =  wL  calories. 

Let  u  be  the  volume  occupied  by  a  gramme  of  the  compound, 
at  the  pressure  P  and  temperature  6,  v  the  volume  of  a  gramme 
of  the  gaseous  component,  v'  the  volume  of  a  gramme  of  the  solid 
component;  the  formation  of  w  grammes  of  the  compound  is 
accompanied  by  an  increase  in  volume 

y  ==  uw — pv — p'vrf 
and  equation  (1)  may  be  written 

E  wL 

6  pv+p'v'—uw' 

Further,  in  many  cases  one  may  with  sufficient  exactness  re- 
place this  formula  by  a  simpler  approximate  formula.  The  vol- 
ume u  occupied  by  a  gramme  of  the  compound,  and  the  volume 
v'  occupied  by  a  gramme  of  the  solid  component,  are  negligible 
compared  with  the  volume  v  occupied  by  a  gramme  of  the  gaseous 
component,  so  that  for  equation  (6)  may  be  substituted 

_E  wL 
'  '  ~~  6    pv' 

Making  use  of  this  formula  and  of  the  results  of  Debray's  in- 
vestigations, giving  an  approximate  value  for  tan  a,  Peslin  was 
able  as  early  as  1871  to  calculate  the  quantity  of  heat  L  set  free 
by  the  formation  of  a  gramme  of  calcium  carbonate  at  a  tempera- 


MONOVARIANT  SYSTEMS. 


179 


ture  included  between  the  cadmium  and  zinc  boiling-points;   he 
found  this  quantity  of  heat  to  have  the  value 

L=293.3cal. 

Favre  and  Silbermann  have  determined  the  heat  of  formation 
of  calcium  carbonate  at  a  temperature  which  they  have  not  stated; 
they  found 

L= 308.1  cal. 

Much  more  satisfactory  applications  of  Clapeyron's  formula  to 
dissociation  phenomena  have  been  made  by  Bonnefoi.1  They 
treat  of  the  combinations  of  lithium  chloride  and  lithium  bromide 
with  ammonia  and  the  amines;  the  following  reactions  liberate  a 
quantity  of  heat  Q  which  has  been  both  measured  by  thermo- 
chemical  methods  (Q0bs.)  and  calculated  by  Clapeyron's  equation 
from  the  dissociation  tensions  (Qcaic.)- 

The  concordance  is  as  satisfactory  as  could  be  wished. 


Reaction. 

Qobs. 

Qcalc. 

LiCl-f  NH3  =LiCl,  NH,.  . 

11  857 

11  9 

LiCl,  NH3+  NH3  =  LiCl,  2NH3  

11  502 

11  6 

LiCl,  2NH3+  NH3  =  LiCl,  3NH3  

11  096 

11  1 

LiCl,  3NH3+  NH3=LiCl,  4NH3  

8  927 

8  9 

LiBr+NH3=LiBr,  NH3.  . 

13  293 

13  37 

LiBr,  NH3+NH3=LiBr,  2NH3  

12  644 

12  74 

LiBr,  2NH3+  NH3  =LiBr,  3NH3  

11  526 

11  51 

LiBr,  3NH3+NH3=LiBr,  4NH3  

10  635 

10  51 

LiCl+  NH2CH,  -LiCl,  XH2CHS  . 

13  815 

13  83 

LiCl,  NH2CH3+  NH2CH,  =  LiCl,  2NH,CH,  . 

12*007 

12  10 

LiCl,  2NH2CH3+  NH.CH3  =  LiCl,  3NH2CH3  

10  870 

10  93 

LiCl+  NH2C2H5  =LiCl,  NH2C2H5 

13  834 

13  71 

LiCl,  NH2C2H5+  NH2C2H-  =LiCl,  2NH,C2H- 

10  983 

11  09 

LiCl,  2NH2C2H5+  NH2C2H5  =LiCl,  3NH2C2H5       . 

10  570 

10  50 

1 J.  BONNEFOI,  Combinaisons  des  sets  haloides  du  lithium  avec  I'amoniac 
et  les  amines  (These  de  Montpellier,  1901). 


CHAPTER  IX. 
MULTIPLE  OR  TRANSITION  POINTS. 

151.  The  same  substance  in  the  three  states:  solid,  liquid, 
gaseous.  Triple  point. — A  great  number  of  substances  of  defi- 
nite composition  may  be  observed,  according  to  circumstances, 
in  the  three  states  of  solid,  liquid,  and  gas;  may  these  three  forms 
coexist  in  equilibrium?  The  system  formed  of  a  single  independent 
component,  divided  into  three  phases,  would"  be  an  invariant 
system;  as  we  know,  there  can  be  but  one  temperature  6  and 
one  pressure  5  for  which  the  system  is  in  equilibrium.  If,  as  in 
the  preceding  chapter,  we  lay  off  on  a  set  of  coordinates  axes 
the  temperatures  T  as  abscissas  and  the  pressures  it  as  ordinates 
(Fig.  44),  a  definite  point,  3,  of  coordinates  6,  *%, 
will  represent  the  conditions  in  which  one  may 
observe  the  substance  in  equilibrium  simultane- 
ously in  the  three  forms  solid,  liquid,  vapor. 
] At  the  temperature  6  and  pressure  ^  the  solid 


O          e       T  anci  the  liquid  may  remain  in  equilibrium  in  con- 

FIG.  44.  tact  with  each  other,  so  that  the  point  3  belongs 

to  the  curve  F  of  fusion  tensions;  the  liquid  may  remain  in 
equilibrium  in  contact  with  the  vapor,  so  that  the  point  3  be- 
longs to  the  curve  V  of  tensions  of  saturated  vapor  from  the 
liquid;  the  solid  may  stay  in  contact  and  in  equilibrium  with  the 
vapor,  so  that  the  point  3  belongs  to  the  curve  V  of  tensions 
of  saturated  vapor  from  the  solid.  Therefore  the  representative 
point  of  the  equilibrium  conditions  of  the  invariant  system  formed 
of  a  given  substance  in  the  three  forms  solid,  liquid,  vapor,  is  a 
point  where  the  three  following  curves  meet: 

1°.  The  curve  of  fusion  tensions; 

2°.  The  curve  of  saturated  vapor  tensions  from  the  liquid; 

3°.  The  curve  of  saturated  vapor  tensions  from  the  solid. 

180 


MULTIPLE  OR  TRANSITION  POINTS. 


181 


From  this  comes  the  name  triple  point  given  to  the  point  3. 

It  may  be  shown  also  that  any  two  of  the  three  curves  of 
which  we  have  just  spoken  can  never  .meet  outside  of  the  triple 
point  3. 

152.  The  transformation  curves  in  the  neighborhood  of  the 
triple  point.  —  It  is  evidently  important  to  know  how  these  three 
curves  are  arranged. 

The  curve  of  fusion  tensions  divides  the  plane  into  two  regions; 
every  point  in  the  region  situated  to  the  right  of  this  curve  repre- 
sents conditions  in  which  the  solid  melts,  while  the  liquid  cannot 
freeze;  every  point  to  the  left  of  this  curve  represents  conditions 
in  which  the  solid  cannot  melt,  while  the  liquid  freezes;  if,  in  this 
region,  the  liquid  is  observed  in  equilibrium,  it  is  because  of  a 
phenomenon  of  false  equilibrium;  the  liquid  is  hi  surfusion. 

At  the  triple  point,  the  two  curves  V  and  V  of  the  tensions 
of  saturated  vapor  cut  the  curve  F  of  fusion  tensions,  so  that  to 
the  left  of  this  curve  there  exists  certainly  a  branch  of  the  curve 
V  and  a  branch  of  the  curve  V]  let  us  see  where  these  branches  lie. 

Can  the  curve  V  exist  above  the  curve  V? 

Suppose  this  to  be  so  and  take  (Fig.  45)  a  point  M  of  abscissa 
T  and  ordinate  nt  situated  between  these    n 
two  curves. 

At  the  pressure  it  and  temperature  T 
the  liquid  freezes,  for  the  point  M  is  to 
the  left  of  the  curve  F  of  fusion  tensions; 
the  solid  vaporizes,  for  the  point  M  is  n 
below  the  curve  V  of  tensions  of  saturated 
vapor  from  the  solid;  the  vapor  con- 
denses  to  the  liquid  state,  for  the  point  M 
is  below  the  curve  V  of  the  tensions  of 
saturated  vapor  from  the  liquid.  One  may  therefore,  at  the 
constant  temperature  T  and  constant  pressure  TT,  put  the  system 
through  a  real  closed  cycle,  which  causes  it  to  pass  from  the  liquid 
to  the  solid  state,  from  the  solid  to  the  vapor,  and  from  the  vapor 
to  the  liquid  state. 

But,  as  the  external  forces  reduce  to  a  constant  pressure,  they 
admit  a  potential  (Chap.  I,  Art.  12),  so  that  the  work  which  they 
perform  about  a  closed  cycle  is  zero.  Our  hypothesis  leads  us 


45« 


182 


THERMODYNAMICS  AND  CHEMISTRY. 


to  regard  as  realizable  a  closed  cycle,  described  at  constant  tem- 
perature, with  kinetic  energy  constantly  zero  and  total  work 
accomplished  equal  to  zero;  we  have  seen  (Chap.  V,  Art.  66)  that 
such  a  cycle  can  be  realized  only  by  the  aid  of  positive  external 
work;  the  hypothesis  assumed  is  therefore  inadmissible,  and  we 
may  state  the  following  theorem: 

To  the  left  of  the  curve  F  of  the  fusion  tensions,  and  consequently 
in  the  conditions  where  the  liquid  is  in  surfusion,  the  curve  V  of  the 
tensions  of  saturated  vapor  from  the  Jiquid  is  above  the  curve  V  of 
the  tensions  of  saturated  vapor  from  the  solid. 

This  theorem  gives  a  correct  idea  of  the  positions  of  the  three 
curves  F,  V,  V. 

There  are  two  cases  to  distinguish: 

FIRST  CASE. — Fusion  is  accompanied  by  an  increase  in  volume. 
This  is  the  case  occurring  the  more  frequently;  acetic  acid  gives 
us  an  often-cited  example.  In  this  case  the  curve  of  fusion  F 
rises  steeply  from  left  to  right;  the  two  curves  V  and  V  rise 
gradually  from  left  to  right;  the  three  curves  appear  as  shown  in 
Fig.  46. 


n 


n 


FIG.  46. 


e 
FIG.  47. 


SECOND  CASE. — Fusion  is  accompanied  by  a  decrease  in  volume. 
This  is  the  case  with  the  fusion  of  ice.  Here  the  fusion  curve 
F  rises  abruptly  from  right  to  left;  the  two  curves  V  and  V  be- 
have as  in  the  preceding  example;  Fig.  47  shows  their  arrangement. 

153.  Historical. — Most  careful  observations  were  for  a  long 
time  powerless  to  detect  a  difference  between  the  tension  of  satu- 
rated vapor  from  an  undercooled  liquid  and  the  tension  of  satu- 
rated vapor  from  the  solid  formed  by  its  freezing, 

We  owe  to  Regnault  a  series  of  Researches  undertaken  to  decide 


MULTIPLE  OR  TRANSITION  POINTS.  183 

if  the  solid  or  liquid  state  of  substances  exercises  an  influence  on  the 
elastic  force  of  the  vapors  that  they  give  off  in  vacuo  at  the  same  tem- 
perature. The  study  of  water,  acetic  acid,  hydrocarbide  of  bro- 
mine and  benzene,  led  the  eminent  physicist  to  the  following 
conclusion:  "We  must  therefore  admit  that  the  molecular  forces 
which  determine  the  solidification  of  a  substance  exercise  no 
sensible  influence  upon  its  vapor  tension  hi  vacuo;  or,  more  ex- 
actly, if  an  influence  of  this  nature  exist,  the  variations  produced 
are  so  small  that  they  cannot  be  detected  in  a  certain  manner 
from  my  experiments." 

"To  sum  up,  my  experiments  prove  that  the  passage  of  a  body 
from  the  solid  to  the  liquid  state  produces  no  appreciable  change 
in  the  curves  of  the  elastic  forces  of  its  vapor;  this  curve  conserves 
a  perfect  regularity  before  and  after  the  transformation." 

As  early  as  1858  this  conclusion  was  attacked,  in  the  name  of 
thermodynamics,  by  G.  Kirchoff,  and  it  is  to  the  investigations  of 
theorists  such  as  Kirchoff,  James  Thomson,  and  J.  Moutier  that  we 
are  indebted  to-day  for  cur  knowledge  of  the  properties  of  vapors 
emitted  by  a  given  substance  in  the  two  states  solid  and  liquid. 

154.  Experimental  verifications. — But  if  experiment  did  not 
precede  theory,  the  latter  once  established  has  received  precise 
confirmations  from  experiment. 

Here  is  an  experiment  due  to  Gernez  which  shows  that  the 
tension  of  saturated  vapor  from  an  undercooled  liquid  exceeds 
the  tension  of  saturated  vapor  of  the  same  substance,  taken  at 
the  same  temperature,  but  hi  the  solid  state. 

An  inverted  U  tube,  whose  ends  are  sealed  off  (Fig.  48)  con- 
tains in  the  branch  A  liquid  acetic  acid,  and  in  the  tube  B  crys- 
tallized acetic  acid;  the  whole  is  kept  at  a  very  nearly  constant 
temperature  by  packing  in  sawdust;  in  these  conditions 
the  liquid  acetic  acid  is  in  a  state  of  surfusion;  the  ten- 
sion of  saturated  vapor  from  the  liquid  is  greater  than 
that  from  the  crystallized  acetic  acid;  therefore,  little  by 
little,  a  distillation  is  established  from  the  branch  A  to 
the  branch  B;  after  several  months  the  liquid  mass  has 
considerably  diminished  in  the  branch  A,  while  in  the  A^ 
branch  B  the  sides  of  the  tube  are  carpeted  with  crystals.  FIG.  48. 

But  we  may  go  farther  and  give  a  quantitative  verification  to 


184  THERMODYNAMICS  AND  CHEMISTRY. 

the  theory;  by  methods  of  great  accuracy,  Lord  Ramsay  and 
Young  on  the  one  hand  and  S.  V.  Fischer  on  the  other,  have 
measured,  below  0°,  the  tension  of  water  vapor  from  ice  and  that 
from  undercooled  water;  the  second  exceeds  the  first,  but  by  a 
very  small  quantity,  0.155  millimetres  of  mercury  at  —  5°  and 
0.22  mm.  at  —10°.  Nevertheless  these  measurements  agree  very 
exactly  with  the  theorems  stated  by  Moutier  and  G.  Kirchoff; 
the  same  is  true  for  the  measurements  of  vapor  tensions  of  ben- 
zene in  the  liquid  and  solid  states,  as  studied  by  Fischer. 

From  the  computations  of  Robert  von  Helmholtz,  the  tem- 
perature of  the  triple  point  for  water  would  barely  exceed  the 
temperature  of  fusion  under  atmospheric  pressure;  it  would  be 
+0°.0076  C.;  the  tension  of  saturated  water  vapor  above  the 
liquid  would  surpass  that  above  ice  by  0.000332  mm.;  on  ac- 
count of  their  smallness  these  numbers  escape  any  exact  experi- 
mental verification. 

155.  Modifications  of  phosphorus.  Researches  of  Troost  and 
Hautefeuille. — We  have  just  studied  substances  taken  at  a  tem- 
perature near  to  the  triple  point;  the  tension  of  saturated  vapor 
from  the  solid  arid  the  tension  of  saturated  vapor  from  the 
undercooled  liquid,  equal  to  each  other  at  the  triple  point  alone, 
differ  only  very  slightly  from  each  other;  only  by  the  most  skil- 
fully arranged  experiments  could  one  detect  this  difference. 

If  we  could  study  the  tensions  of  saturated  vapor  from  the 
undercooled  liquid  and  from  the  solid  which  arises  from  freezing 
at  a  temperature  widely  separated  from  the  triple  point,  it  is 
to  be  presumed  that  the  two  tensions  of  saturated  vapor  will 
be  extremely  different  and  that  even  rough  experiments  would 
be  sufficient  to  distinguish  them.  The  vaporization  of  liquid 
white  phosphorus  and  of  solid  red  phosphorus  will  give  us  an 
example  of  the  vaporization  of  a  single  substance  in  two 
different  states,  at  temperatures  very  distant  from  the  triple 
point. 

White  solid  phosphorus  at  ordinary  temperatures  and  white 
liquid  phosphorus  above  its  fusing-point  are  transformed  slowly 
into  red  solid  phosphorus  under  the  action  of  light;  at  higher 
temperatures  the  transformation  becomes  more  and  more  rapid 
and  takes  place  even  in  the  dark;  the  inverse  transformation  of 


MULTIPLE  OR  TRANSITION  POINTS.  185 

red  phosphorus  into  white  does  not  occur  at  any  temperature 
attainable  in  the  laboratory;  the  curve  of  transformation  tensions 
of  red  into  white  phosphorus,  if  it  exists,  is  beyond  the  field  of 
observable  temperatures. 

The  transformation  of  liquid  white  phosphorus  into  red  takes 
place  with  liberation  of  heat;  thus,  according  to  Hittorf,  when  at 
280°  liquid  white  phosphorus  is  rapidly  transformed  into  solid  red 
phosphorus,  the  heat  set  free  is  so  great  that  the  temperature  is 
suddenly  raised  from  280°  to  370°  C. 

The  transformation  of  liquid  white  into  solid  red  phosphorus 
takes  place  also  with  decrease  in  volume. 

At  temperatures  attainable  hi  the  laboratory  we  may  observe, 
as  has  been  said,  the  exothermic  transformation  of  liquid  white 
phosphorus  into  the  red  solid  form,  but  not  the  inverse  transfor- 
mation; the  region  where  the  observations  are  made  is,  therefore, 
according  to  Moutier's  rule,  to  the  left  of  the  curve  of  tensions 
of  transformation  from  red  phosphorus  into  liquid  white  phos- 
phorus; if  the  transformation  of  red  into  liquid  white  phosphorus 
is  possible,  it  is  at  temperatures  much  higher  than  those  realiza- 
ble in  our  laboratories. 

All  these  properties  allow  us  to  regard  liquid  white  phosphorus 
as  a  liquid  which  may  freeze  in  various  forms:  certain  of  these 
forms,  as  amorphous  or  crystallized  white  phosphorus,  cannot 
exist  at  the  relatively  high  temperatures  which  we  consider;  as  for 
the  form  red  phosphorus,  its  point  of  fusion  into  the  liquid  white 
form,  if  it  exists,  is  very  much  higher  than  the  temperatures  which 
we  can  reach;  at  these  temperatures  the  liquid  white  phosphorus 
should  be  considered  as  a  liquid  in  surfusion  with  respect  to  the  solid 
red  phosphorus. 

At  any  given  temperature  the  tension  of  saturated  vapor  of 
liquid  white  phosphorus  should  exceed  the  tension  of  saturated 
vapor  of  red  phosphorus. 

Thus  if  liquid  white  phosphorus  is  heated  to  440°,  for  exam- 
ple, as  was  done  by  Troost  and  Hautefeuille,1  the  vapor  pressure 
attains  quickly  7.75  atmos.,  which  is,  at  this  temperature,  the 

1  TROOST  and  HAUTEFEUILLE,  Annales  de  VEcole  superieure,  2d  Series, 
v.  2,  p.  253,  1873. 


186 


THERMODYNAMICS  AND  CHEMISTRY. 


tension  of  saturated  vapor  from  liquid  white  phosphorus;  but 
the  white  phosphorus  is  gradually  changed  into  red,  and  when 
this  transformation  is  complete  the  tension  of  the  phosphorus 
vapor  is  lowered  to  1.75  atmospheres. 

If  red  phosphorus  is  heated  to  440°,  the  vapor  pressure  in- 
creases slowly  up  to  1.75  at.  and  remains  stationary  when  this 
value  has  been  reached. 

Troost  and  Hautefeuille  denoted  the  pressure  of  1.75  at.  as 
being  at  440°  the  transformation  tension  of  phosphorus  vapor; 
from  what  precedes  we  see  that  this  pressure  should  be  considered 
simply  as  the  tension  of  saturated  vapor  from  red  phosphorus. 

Troost  and  Hautefeuille  determined  at  various  temperatures 
the  tensions  of  saturated  vapor  from  white  liquid  phosphorus 
and  from  red  phosphorus;  at  the  same  temperature  the  first  ten- 
sion is  always  very  considerably  higher  than  the  second.  Below 
are  the  results  of  these  measurements: 


Temperature. 

Tension  of  Saturated  Vapor. 

From  White 
Phosphorus. 

From  Red 
Phosphorus. 

360°  C. 
440 
487 
494 
503 
510 
521 
531 
550 
577 

3.  20  at. 

7.75 

18.00 
21.90 

0.12  at. 
1.75 
6.80 

10.8 

16.0 
31.0 
56.0 

26.20 

Above  510°  the  transformation  of  liquid  white  phosphorus  into 
red  phosphorus  is  so  rapid  that  the  tension  of  saturated  vapor 
from  white  phosphorus  has  not  the  time  to  become  established 
and  can  no  longer  be  measured. 

156.  Researches  of  G.  Lemoine. — When  phosphorus  vapor 
is  brought  to  a  high  temperature  and  then  suddenly  cooled  to 
room  temperature,  it  condenses  in  the  form  of  white  phos- 


MULTIPLE  OR  TRANSITION  POINTS.  187 

phorus;  this  fact  may  be  of  use  in  explaining  Lemoine's  observa- 
tions.1 

If  we  enclose  in  a  flask  a  mass  of  white  or  red  phosphorus, 
and  if  we  keep  this  flask  at  a  fixed  high  temperature  T  long 
enough  for  equilibrium  to  be  established,  we  shall  have,  at  the 
end  of  the  experiment,  the  flask  filled  with  phosphorus  vapor, 
whose  tension  will  be  equal  to  the  tension  of  saturated  vapor 
from  red  phosphorus  at  the  temperature  of  the  experiment,  and 
the  mass  of  solid  in  excess  will  be  in  the  form  of  red  phos- 
phorus. If,  for  instance,  the  temperature  is  440°  C.,  we  shall 
have  a  mass  of  phosphorus  vapor  which  will  fill  the  flask  at  a 
pressure  of  1.75  at.;  this  mass  will  be  equal  to  as  many  times 
3.6  gr.  as  there  are  litres  in  the  volume  of  the  flask;  the  rest  of 
the  phosphorus  will  be  in  the  red  form.  By  suddenly  cooling  the 
flask  we  shall  find  a  mass  of  phosphorus,  soluble  in  carbon  bisul- 
phide, equal  to  3.6  gr.  per  litre;  the  rest  will  be  insoluble  in  carbon 
bisulphide. 

Lemoine  was  not  content  to  study  the  state  of  equilibrium 
which  is  set  up  in  such  a  flask  after  a  continued  heating;  by  sub- 
mitting such  flasks  to  a  sudden  cooling  after  variable  times  of 
heating,  he  studied  the  way  in  which  this  equilibrium  is  estab- 
lished when  starting  either  with  white  or  with'  red  phosphorus. 

In  the  first  case  the  mass  of  white  phosphorus  which  the  flask 
holds  should  constantly  decrease  up  to  the  limiting  value,  which 
is  3.6  gr.  per  litre;  in  the  second  case  this  quantity  should  in- 
crease, reaching,  without  ever  exceeding,  the  same  limit. 

157.  Anomaly  observed  by  Lemoine. — The  results  of  Le- 
moine's  observations  always  accord  with  prediction  in  the  first- 
case,  but  not  always  in  the  second;  if  a  considerable  quantity  of 
red  phosphorus  is  heated  in  a  closed  vessel,  it  is  noted  that  the 
mass  of  white  phosphorus  coming  from  the  condensation  of  the 
phosphorus  vapor  increases  at  first  so  as  to  exceed  3.6  gr.  per 
litre,  passes  through  a  maximum,  then  decreases  to  3.6  gr. 

For  example,  here  are  the  results  of  experiments  where  Le- 
moine heated  30  gr.  of  red  phosphorus  to  440°  in  flasks  of  1  litre; 


1  G.  LEMOINE,  Annales  de  Chimie  et  de  Physique,  4th  Series,  v.  24,  p.  129, 
1871. 


188 


THERMODYNAMICS  AND   CHEMISTRY. 


the  masses  of  white  phosphorus  obtained  after  variable  times 
of  heating  are  the  following: 


Time  in  Hours. 

Masses  of  White 
Phosphorus. 

Time  in  Hours. 

Masses  of  White 
Phosphorus. 

0  h.  30  min. 
2  h. 
8  h. 

4.54gr. 
4.75 
4.40 

23  h. 
32 
47 

3.90gr. 
3.74 
3.72 

158.  Explanation  of  this  anomaly  by  Troost  and  Hautefeuille, 

— The  explanation  of  this  anomaly  has  been  given  by  Troost  and 
Hautefeuille.1 

There  exists  not  merely  one,  but  a  great  number  of  varieties 
of  red  phosphorus;  the  properties  of  red  phosphorus  depend 
upon  the  temperature  at  which  it  is  produced;  prepared  at  265° 
it  is  a  bright  red  having  brilliant  vitreous  breaks,  suggesting  by 
its  brightness  realgar;  red  phosphorus  obtained  at  440°  is  orange, 
its  breaks  are  dull  and  granular;  obtained  above  500°  it  is  com- 
pact, has  a  bright  violet  hue,  a  conchoid  break,  and  in  its  cavities 
are  formed  geodes  of  red  phosphorus  crystals. 

These  various  kinds  differ  in  density  and  heat  of  combustion. 

Let  us  take  for  standard  of  comparison  crystallized  red  phos- 
phorus, whose  density  is  2.34. 

Commercial  red  phosphorus  has  a  heat  of  combustion  which 
exceeds  by  568  calories  per  gramme  that  of  crystallized  red  phos- 
phorus. 

Red  phosphorus  kept  at  265°  for  650  hours  has  a  density  of 
2.148;  its  heat  of  combustion  exceeds  by  320  calories  per  gramme 
that  of  crystallized  red  phosphorus. 

Kept  540  hours  at  360°  the  density  of  red  phosphorus  is  2.19, 
and  its  heat  of  combustion  exceeds  by  298  calories  that  of  red 
crystallized  phosphorus. 

Phosphorus  prepared  at  580°  has  a  heat  of  combustion  of  50 
calories  less  than  that  of  crystallized  red  phosphorus. 


1  TROOST  and  HAUTEFEUILLE,  Annales  de  Chimie  et  de  Physique,  5th  S., 
v.  2,  p.  155,  1874. 


MULTIPLE  OR   TRANSITION  POINTS.  189 

"Red  phosphorus  does  not  assume  immediately  the  peculiar 
appearance  that  we  have  pointed  out;  this  is  acquired  slowly  if 
the  experiment  is  performed  at  a  moderately  high  temperature, 
and  very  rapidly  if  above  500°." 

"The  varieties  which  cease  to  be  modified  by  a  new  heating 
for  a  great  number  of  hours  at  the  same  temperature,  pass  from 
one  form  to  the  other  by  insensible  changes  when  they  are  brought 
to  a  higher  temperature  kept  a  long  time  constant." 

We  see  by  wrhat  precedes  that  the  varieties  obtained  at  low 
temperatures  should  be  regarded  as  existing  in  the  state  of  false 
equilibrium,  when  they  are  brought  to  a  higher  temperature; 
they  change  into  the  variety  corresponding  to  this  temperature; 
the  reasoning  given  in  Art.  152  may  therefore  be  applied  here  and 
shows  that  the  first  varieties  have  a  tension  of  saturated  vapor 
higher  than  the  last;  Troost  and  Hautefeuille  have  in  fact  proved 
that  at  a  given  temperature  the  various  forms  of  red  phosphorus 
have  a  tension  which  is  the  higher  in  proportion  as  they  are  pre- 
pared at  a  lower  temperature;  thus  red  phosphorus  obtained 
at  265°  should  behave  at  440°,  with  respect  to  the  red  phosphorus 
obtained  at  this  latter  temperature,  as  white  phosphorus  acts 
with  respect  to  red  phosphorus;  this  is  what  Troost  and  Haute- 
feuille have  shown. 

Lemoine  having  worked  at  440°  with  commercial  red  phos- 
phorus prepared  between  250°  and  300°,  the  vapor  tension  should 
have,  as  in  Troost  and  Hautefeuille's  experiments,  increased 
rapidly  up  to  the  tension  of  saturated  vapor  from  commercial 
red  phosphorus,  then  decrease  to  the  tension  of  saturated  vapor 
of  red  phosphorus  prepared  at  440°. 

159.  The  triple  point  considered  as  transition  point. — Con- 
sider a  system  that  may  contain  water  in  the  three  forms  liquid, 
ice,  and  vapor;  at  a  temperature  differing  from  the  temperature 
6  of  the  triple  point  3  the  system  cannot  be  in  equilibrium  unless 
at  least  one  of  the  three  phases  has  disappeared. 

Three  kinds  of  systems  formed  of  two  phases  may  be  observed: 
the  system  made  of  ice  and  water,  that  of  ice  and  vapor,  and  the 
one  formed  of  water  and  vapor. 

For  the  first  system  to  be  in  equilibrium,  the  point  representing 


190  THERMODYNAMICS  AND  CHEMISTRY. 

this  state  must  be  on  the  line  FF^  (Fig.  49);  in  order  that  the 
second  system  may  be  in  equilibrium 
it  is  necessary  that  the  point  representing 
it  lie  on  the  line  F'F/;  and  for  the  third 
to  be  in  equilibrium  its  representative 
point  must  lie  on  FFj. 

These    conditions    are   not   in   general 
sufficient  to  assure  equilibrium. 

o  Q  Consider,  at  a  temperature  less  than 

FIG.  49.  6,  a  system  formed  of    liquid  water  and 

vapor  and  suppose  that  the  representative  point  is  on  the  curve 
FtO  of  tensions  of  saturated  vapor  from  liquid  water;  will  the 
system  be  in  equilibrium?  No,  for  the  vapor,  whose  tension  is 
greater  than  that  of  saturated  vapor  from  ice  at  the  same  tem- 
perature, may  condense  into  the  form  ice;  the  liquid,  whose  tem- 
perature is  less  than  the  fusing-point  for  the  same  pressure,  may 
freeze.  The  system  formed  of  liquid  water  and  vapor  is  there- 
fore not  in  equilibrium  in  the  conditions  which  we  have  described, 
or  at  least,  if  it  may  be  observed  in  equilibrium,  it  is  one  of  those 
false  equilibria  .to  which  we  have  called  attention;  it  is  through  ' 
these  false  equilibria  that  we  can  trace,  in  certain  cases,  the  line 
FjO  of  tensions  of  saturated  vapor  from  water  in  surfusion. 

Take  next,  at  a  temperature  above  6,  a  system  formed  of  ice 
and  vapor  and  suppose  that  the  representative  point  is  on  the 
curve  OF/  of  tensions  of  saturated  vapor  from  ice;  the  repre- 
sentative point  being  above  the  curve  OF  of  tensions  of  saturated 
vapor  from  liquid  water,  the  vapor  may  condense  as  liquid;  the 
representative  point  being  to  the  right  of  the  curve  of  fusion  points 
FFlt  the  ice  may  melt;  therefore  the  system  is  not  in  equilibrium. 
Take,  finally,  a  system  formed  of  ice  and  liquid  at  a  tempera- 
ture higher  than  6,  and  suppose  that  the  representative  point 
is  on  the  curve  OFt  of  the  points  of  fusion;  the  representative 
point  being  below  the  two  curves  of  tensions  of  saturated  vapor, 
the  liquid  water  and  the  ice  may  evaporate  and  the  system  is  not 
in  equilibrium. 

In  short,  at  temperatures  below  6  two  kinds  of  true  equilib- 
rium may  be  observed: 

1°.  Systems  consisting  of  ice  and  water  vapor;   it  is  necessary 


MULTIPLE  OR   TRANSITION  POINTS. 


191 


and  sufficient  that  the  representative  point  be  on  the  branch 
F'3  (indicated  by  a  full  line)  of  the  curve  of  tensions  of  saturated 
vapor  from  ice. 

2°.  Systems  formed  of  ice  and  water;  it  is  necessary  and  suffi- 
cient that  the  representative  point  be  on  the  branch  F3  (indi- 
cated by  a  full  line)  of  the  curve  of  fusing-points. 

At  temperatures  above  6  there  may  be  had  a  single  kind  of 
system  in  the  state  of  true  equilibrium,  namely,  systems  formed 
of  liquid  water  and  vapor;  it  is  necessary  and  sufficient  that  the 
representative  point  be  on  the  branch  3F  (indicated  by  a  full 
line)  of  the  curve  of  tensions  of  saturated  vapor  from  liquid  water. 

A  similar  treatment  may  be  given  to  a  system  where  a  single 
substance  may  exist  in  the  three  forms  solid,  liquid,  vapor,  and 
when  the  fusion  is  accompanied  by  an  increase  in  volume.  In  this 
case  the  curve  of  fusing-points  FJ?  (Fig.  50)  rises  from  left  to  right. 

The  results  attained  are  the  following : 

At  temperatures  less  than  the  temperature  6  of  the  triple 
point  there  may  be  observed  in  equi- 
librium only  a  single  kind  of  mono- 
variant  system,  that  formed  of  solid 
and  vapor;  in  order  for  this  equilibrium 
to  take  place  it  is  necessary  and  suffi- 
cient that  the  representative  point  be 
on  the  branch  V'3  of  the  curve  of  ten- 
sions of  saturated  vapor  from  the  solid. 

At  temperatures  higher  than  the 
temperature  0  of  the  triple  point  two 
kinds  of  monovariant  systems  in  equilibrium  may  be  observed: 

1°.  Systems  formed  of  liquid  and  vapor;  it  is  necessary  and 
sufficient  for  this  that  the  representative  point  be  on  the  branch 
3F  of  the  curve  of  tensions  of  saturated  vapor  from  the  liquid. 

2°.  Systems  formed  of  solid  and  liquid;  it  is  necessary  and 
sufficient  that  the  representative  point  lie  on  the  branch  3F  of 
the  curve  of  fusing-points. 

Let  us  generalize  the  observations  which  we  have  made  foi 
the  two  cases  we  have  treated: 

When  a  single  substance  may  exist  in  three  different  states, 
these  three  states  united  form  an  invariant  system;  this  system 


e 
FIG.  50. 


192  THERMODYNAMICS  AND  CHEMISTRY. 

cannot  be  in  equilibrium  except  at  a  single  temperature  6  and  at 
a  single  pressure  ^,  temperature  and  pressure  of  the  triple  point. 

By  grouping  these  states  two  by  two,  three  monovariant  sys- 
tems are  obtained;  each  of  these  systems  can  be  in  equilibrium 
only  if  the  temperature  is  to  a  well  determined  side  of  the  tem- 
perature 0  of  the  triple  point;  some  can  be  in  equilibrium  only 
at  temperatures  less  than  6;  others  only  at  temperatures  higher 
than  0. 

If,  therefore,  a  monovariant  system  is  taken  in  equilibrium 
and  the  temperature  is  so  varied  that  it  reaches  the  temperature 
0  of  the  triple  point,  at  the  instant  this  value  is  passed,  the  mono- 
variant  system  considered  can  no  longer  be  kept  in  equilibrium; 
it  passes  perforce  into  another  kind  of  system,  whence  the  names 
transition  temperature,  transition  po'ni,  given  by  Bakhuis  Rooz- 
boom  to  the  temperature  0  and  the  triple  point  3. 

160.  Generalization  of  the  preceding  ideas.  Quadruple  points. 
— These  ideas  may  be  generalized. 

Take  a  system  formed  of  two  independent  components  and 
suppose  that  it  may  be  divided  into  four  phases.  If  the  four 
phases  coexist  in  the  system,  it  is  invariant;  there  is  a  single  tem- 
perature 0  and  a  single  pressure  2,  coordinates  of  a  definite  point 
3,  at  which  this  invariant  system  may  be  observed  in  equilibrium. 

If  one  of  the  four  phases  is  supposed  to  be  excluded,  a  mono- 
variant  system  is  obtained;  according  as  the  excluded  phase  is 
one  or  another  of  the  four  possible  phases,  there  may  be  formed 
four  distinct  monovariant  systems. 

In  order  that  one  of  these  four  monovariant  systems  may 
be  in  equilibrium  it  is  necessary  that  the  representative  point 
be  on  the  curve  of  transformations  corresponding  to  this  system; 
to  the  four  possible  monovariant  systems  correspond  four  curves 
of  transformation  tensions. 

It  is  clear  that  each  of  these  four  curves  must  pass  through 
the  point  3,  which  is  therefore  a  quadruple  point. 

To  have  one  of  our  four  systems  in  equilibrium  it  is  necessary 
for  the  point  which  represents  the  temperature  and  pressure  to  be 
on  the  curve  of  transformation  tensions  of  this  system;  but  this 
does  not  suffice;  the  curve  of  transformation  tensions  does  not 
represent,  in  its  entirety,  states  of  equilibrium;  only  one  branch 


MULTIPLE  OR   TRANSITION  POINTS.  193 

of  this  curve  should  be  kept,  and  this  branch  has  the  point  3 
for  an  end,  so  that  the  quadruple  point  is  a  transition  point. 

161.  Quadruple  points  in  the  study  of  hydrates  from  a  gas. — 
Bakhuis  Roozboom  and,  after  him,  other  chemists  have  studied 
a  certain  number  of  quadruple  points;   the  first  quadruple  points 
which  have  been  discussed  are  found  in  the  study  of  systems 
whose  two  independent  components  are  water  and  a  gas  and  which 
may  have  four  phases: 

Ice; 

Solid  hydrate; 
Liquid  mixture; 
Gaseous  mixture. 

162.  Quintuple   points. — We   may  go   farther;   in   a   system 
formed  of  three  independent  components,  susceptible  of  having 
five  phases,  a  quintuple  point  will  be  met,  where  the  five  curves 
of  transformation  will  cut  which  correspond  to  the  five  mono- 
variant  systems  obtained  by  excluding  successively  one  of  the 
five  phases;   this  quintuple  point  is  a  transition  point. 

Here  are  two  examples  of  quintuple  points  analyzed  by 
Bakhuis  Roozboom: 

Take  a  system  formed  of  three  independent  components:: 
Water:  H,0; 

Sodium  sulphate :  Na2S04; 
Magnesium  sulphate :  MgS04. 

At  a  temperature  of  22°  the  following  five  phases  may  co- 
exist: 

1°.  Water  vapor,  7; 

2°.  A  liquid  mixture  L,  formed  of  water,  sodium  sulphate,  and 
magnesium  sulphate; 

3°.  Crystals  N  of  hydrated  sodium  sulphate:  Na2S04-10H2O; 
4°.  Crystals  M  of  hydrated  magnesium  sulphate :  MgSO4  •  10H2O ; 
5°.  Crystals  A  of  astrakanite,  whose  formula  is 

Na2MgS2O8-4H20. 

The  temperature,  0=22°,  is  that  of  the  quintuple  point  3;  in 
the  neighborhood  of  this  point  the  five  curves  of  transformation 
are  arranged  as  is  shown  in  Fig.  51. 

Take  a  system  whose  independent  components  are' 


194 


THERMODYNAMICS  AND  CHEMISTRY. 


Water; 

Cupric  acetate:  Cu(C2H302)2; 

Calcic  acetate:    Ca(C2H302)2. 
At  a  temperature  of  76°  there  may  coexist] 
1°.  Water  vapor,  7; 


n 


n 


o 


T    o 


FIG.  51. 


O 
FIG.  52. 


2°.  A  liquid  mixture  L,  formed  of  water,  cupric  acetate,  and 
calcium  acetate; 

3°.  Crystals  Ct  of  hydrated  cupric  acetate:  Cu(C2H3O2)-H2O; 

4°.  Crystals  C2  of  hydrated  calcic  acetate:  Ca(C2H3O2)-H2O; 

5°.  Crystals  D  of  cupri-calcic  acetate    CuCa(C2H302)2-6H2O. 

The  temperature  76°  is  the  temperature  of  a  quintuple  point 
3;  in  the  neighborhood  of  this  point  the  five  curves  of  trans- 
formation tensions  are  arranged  as  shown  in  Fig.  52. 


CHAPTER  X. 
THE  DISPLACEMENT  OF  EQUILIBRIUM. 

163.  In  general  a  modification  which  leaves  invariable  the 
composition  of  each  phase  cannot  be  imposed  upon  a  system 
whose  variance  exceeds  i. — We  have  seen  (Chap.  VIII,  Art.  145) 
that  we  could  always  impose  upon  a  monovariant  system  a  modi- 
fication which,  while  changing  the  mass  of  the  different  phases, 
left  in  variable  the  composition  of  each;  we  concluded  from  this 
that  if  we  took  a  monovariant  system  in  equilibrium  at  a  given 
temperature  and  under  the  transformation  tension  correspond- 
ing to  this  temperature,  and  if,  moreover,  the  temperature  and 
pressure  were  kept  constant,  the  equilibrium  of  the  monovariant 
system  was  indifferent. 

These  properties  no  longer  hold  when  we  study  systems  whose 
variance  exceeds  unity  or,  at  least,  are  only  met  with  in  certain 
particular  cases ;  thus  in  the  next  chapter  we  shall  see,  in  studying 
bi variant  systems,  that  such  a  system  may  sometimes  have  a 
particular  equilibrium  state  where  are  found  all  the  properties 
which  the  equilibrium  states  of  a  monovariant  system  have  mani- 
fested; but  even  in  the  case  where  such  a  state  of  equilibrium 
will  be  possible  it  will  represent  an  exception  among  the  cases 
of  equilibrium  of  the  system  considered. 

In  general,  for  a  bivariant  system,  every  modification  which 
alters  the  mass  of  the  phases  alters  at  the  same  tune  the  com- 
position of  some  among  them. 

Take,  for  instance,  a  bivariant  system  consisting  of  two  inde- 
pendent components,  water  and  sodium  chloride,  divided  into  two 
phases,  solid  sodium  chloride  and  an  aqueous  solution  of  this 
salt;  if  we  seek  to  increase  the  mass  of  the  liquid  phase  and  de- 

195 


196  THERMODYNAMICS  AND  CHEMISTRY. 

crease  the  mass  of  the  solid  phase,  we  shall  necessarily  decrease 
the  concentration  of  the  solution. 

164.  In  general  the  equilibrium  of  a  system  whose  variance 
exceeds  i  is  stable. — Suppose  given  a  pressure  and  temperature 
and  let  us  keep  them  constant.    The  bivariant  system  studied 
can  be  put  into  equilibrium  at  this  temperature  and  pressure; 
for  this  it  is  necessary  that  each  phase  composing  the  system  has 
a  definite  composition;    take  such  a  state  of  equilibrium,  and, 
starting  with  this  state,  subject  the  system  to  a  slight  modifica- 
tion, which  causes  the  masses  of  one  or  of  some  of  the  phases  to 
vary;   this  modification  changes  the  composition  of  at  least  one 
of  the  phases  into  which  the  system  is  divided;   the  system  which 
was  in  equilibrium  before  the  modification  can  no   longer  be  so 
after  this  change;  the  original  equilibrium  state  was  therefore  not 
a  state  of  indifferent  equilibrium. 

If  the  temperature  and  pressure  are  kept  constant,  this  state 
of  equilibrium  is  stable;  when  it  has  been  disturbed  by  a  small 
modification  similar  to  that  we  have  spoken  of,  the  system  under- 
goes spontaneously  a  modification  in  the  opposite  direction. 

Take,  for  example,  one  system  formed  of  solid  sodium  chloride 
and  an  aqueous  solution  of  this  salt;  suppose  it  in  equilibrium  at 
a  given  pressure  and  temperature;  under  these  conditions  the 
solution  is  saturated  with  sodium  chloride;  without  changing  the 
temperature  or  the  pressure,  imagine  that  any  cause  brings  into 
the  solution  a  small  quantity  of  salt;  immediately  the  solution, 
become  supersaturated,  will  deposit  sodium  chloride  and  return 
to  its  original  concentration;  imagine,  on  the  contrary,  that  the 
saturated  solution  deposits  a  small  quantity  of  sodium  chloride; 
it  will  cease  to  be  saturated  and  will  dissolve  salt  until  it  has 
returned  to  its  original  composition. 

What  we  have  just  said  regarding  a  bivariant  system  may 
be  repeated  for  any  multi variant  system  whatever:  //  certain 
exceptional  equilibrium  states  which  are  indifferent  are  excluded, 
every  state  of  equilibrium  of  a  bivariant  or  multivariant  system  is 
stable  when  the  temperature  and  pressure  are  kept  constant. 

165.  Displacement  of  equilibrium  by  variation  of  the  pressure. 
— From  the  fact  that  the  equilibrium  of  a  bivariant  or  multi- 
variant  system  is  stable  when  the  pressure  and  temperature  are 


THE  DISPLACEMENT  OF  EQUILIBRIUM.  197 

constant  there  follow  two  laws  of  great  importance :  the  law  of  the 
displacement  of  equilibrium  by  variation  of  the  pressure  and  the  law 
of  the  displacement  of  equilibrium  by  variation  of  the  temperature. 

Consider  the  first  of  these  laws,  which  may  be  stated  in  the 
following  way: 

Take  a  system  in  stable  equilibrium  at  a  given  temperature  and 
pressure;  without  changing  the  temperature,  INCREASE  THE  PRESSURE 
by  a  small  amount;  in  general  the  equilibrium  will  be  disturbed; 
the  system  will  be  the  seat  of  a  small  reaction  which  will  bring  it  into 
a  new  state  of  equilibrium;  if  the  same  reaction  is  supposed  to  take 
place,  starting  from  the  initial  equilibrium  state,  without  change  of 
pressure  or  temperature,  it  would  be  accompanied  by  a  DECREASE 
IN  VOLUME  of  the  system. 

If  we  had  disturbed  the  initial  equilibrium  by  DECREASING  slightly 
THE  PRESSURE,  we  should  have  produced*  a  small  reaction  in  the 
system;  taking  place  in  the  initial  state  of  equilibrium,  without 
change  of  temperature  or  pressure,  this  reaction  would  have  been 
<Lccompanied  by  an  INCREASE  IN  VOLUME  of  the  system. 

1 66.  Various  applications. — Here  is  an  example  showing  how 
easy  this  law  is  to  apply: 

A  vessel,  brought  to  1000°  at  a  pressure  TT,  contains  oxygen, 
hydrogen,  and  water  vapor;  let  x  be  the  ratio  between  the  mass 
m  of  the  water  vapor  which  the  system  encloses  and  the  mass  M 
of  this  vapor  which  it  would  enclose  if  the  combination  of  oxygen 
and  hydrogen  were  carried  to  the  disappearance  of  one  of  the  two 
gases.  For  the  system  in  equilibrium  at  the  pressure  n  this  ratio 
x  has  a  certain  value  X. 

Leaving  the  temperature  equal  to  1000°,  cause  the  pressure  to 
take  on  a  value  Ti7  a  little  greater  than  TT;  a  reaction  is  produced 
in  the  system;  the  ratio  x  passes  from  the  value  X  to  the  value  X' 
which  assures  the  equilibrium  at  the  temperature  of  1000°  and 
under  the  pressure  Tr7;  accomplished  at  the  constant  temperature 
1000°  and  under  the  constant  pressure  TT,  this  reaction  will  be 
accompanied  by  a  decrease  in  the  volume  of  the  system. 

Now,  at  constant  temperature  and  pressure,  the  formation  of 
a  certain  quantity  of  water  vapor,  reaction  which  increases  x,  is 
accompanied  by  a  decrease  in  volume;  the  dissociation  of  a  cer- 
tain quantity  of  water  vapor,  reaction  decreasing  x,  is  accom- 


198  THERMODYNAMICS  AND  CHEMISTRY. 

panied  by  an  increase  in  volume;  it  follows,  therefore,  that  X* 
is  greater  than  X. 

Whence  the  following  conclusion: 

At  a  given  temperature,  1000°  for  example,  a  system  is  taken 
which  encloses  a  given  mass  of  oxygen  (free  or  combined)  and  a 
given  mass  of  hydrogen  (free  or  combined) ;  the  ratio  of  the  mass 
of  water  vapor  contained  by  the  system  in  equilibrium  to  the  mass 
of  water  vapor  which  it  would  contain  if  the  combination  was 
complete  is  the  greater  as  the  pressure  is  higher. 

In  other  terms,  at  a  given  temperature  a  decrease  in  pressure 
favors  the  dissociation  of  water,  an  increase  in  pressure  aids  the 
combination  of  oxygen  and  hydrogen. 

When  a  mixture  of  chlorine  and  water  vapor  passes  over  par- 
tially at  constant  temperature  into  oxygen  and  hydrochloric  acid 
the  reaction  is  accompanied  by  an  increase  in  volume;  this  re- 
action is  helped,  therefore,  by  decreasing  the  pressure;  by  increas- 
ing the  pressure  the  inverse  reaction  is  aided. 

167.  Case  of  combinations  without  contraction. — When  iodine 
vapor  and  hydrogen  combine  at  constant  temperature  and  pres- 
sure to  form  hydriodic  acid  the  reaction  is   accompanied  by  no 
change  in  volume,  at  least  in  the  conditions  where  the  iodine  vapor, 
hydrogen,  and  hydriodic  acid  may  be  treated  as  perfect  gases, 
In  reasoning  as  we  have  for  the  system  oxygen,  hydrogen,  water 
vapor,  we  may  draw  from  the  law  of  the  displacement  of  equilib- 
rium by  pressure  the  following  conclusions: 

A  change  in  pressure  without  variation  of  temperature  cannot, 
in  such  a  system,  help  either  the  formation  or  the  dissociation 
of  hydriodic  acid;  if  a  system  of  definite  elementary  composition 
is  taken  and  heated  to  a  definite  temperature,  the  ratio  Y  be- 
tween the  mass  of  free  hydrogen  and  the  total  hydrogen,  free  or 
combined,  which  it  encloses  will  have  a  value  independent  of 
the  pressure  supported  by  the  system. 

168.  Experimental  verifications:  hydriodic  acid. — This  propo- 
sition has  been  verified  by  G.  Lemoine.1 

A  system  containing  hydrogen  and  iodine  vapor  in  equivalent 

1  G.  LEMOINE,  Annaks  de  Chimie  et  de  Physique,  5th  Series,  t.  12,  p.  145, 
1877. 


THE  DISPLACEMENT  OF  EQUILIBRIUM. 


199 


proportions  is  brought  to  the  sulphur  boiling-point  and  subjected 
to  various  pressures;  to  each  value  it  of  the  pressure  corresponds 
a  value  of  Y  as  follows : 


JT 

Y 

jr 

Y 

4.  5  at. 

0.24 

0.9  at. 

0.26 

2.3 

0.25 

0.2 

0.29 

In  this  series  of  experiments,  while  the  pressure  has  passed 
from  one  value  to  another  twenty-two  times  smaller,  the  ratio  F 
has  changed  by  only  ^  of  its  value;  if  account  is  taken  of  the 
fact  that  the  substances  are  not  perfect  gases,  of  the  difficulty  of 
the  experiments  and  the  numerous  sources  of  error,  such  a  con- 
clusion appears  justified. 

169.  Selenic  acid. — The  formation  of  gaseous  selenic  acid 
from  liquid  selenium  and  hydrogen  is  also  a  reaction  which,  taking 
place  at  constant  temperature  and  pressure,  produces  but  a  very 
slight  change  in  volume.  Take  now  a  system  where  hydrogen 
and  gaseous  selenic  acid  exist  in  presence  of  liquid  selenium;  the 
system  is  in  equilibrium  at  a  given  temperature  at  the  pressure  TT; 
it  contains  a  mass  m  of  selenic  acid;  if  the  whole  of  the  hydrogen 
passed  into  the  form  of  selenic  acid,  it  would  enclose  a  mass  M ; 

the  ratio  -TF  has  a  certain  value  X;  if,  without  changing  the  tem- 
perature, the  value  TT  of  the  pressure  is  changed,  the  value  of  X 
should  undergo  only  small  variations. 

Ditte  *  tried  to  verify  this  proposition  experimentally  ;  Pela- 
bon  2  made  a  more  careful  verification ;  here  are  the  results  of 
his  observations: 

TEMPERATURE  620°.  The  total  pressure  of  the  gaseous  mix- 
ture reduced  to  23°  was  520  mm.  of  mercury: 

Z=0.4067. 

1  DITTE,  Annales  de  VEcole  normale  superieure,  2d  Series,  v.  I,  p.  293, 
1873. 

2  H.  PELABON,  Mem.  de  la  Societe  des  Sciences  physiques  et  naturettes  de 
Bordeaux,  5th  Series,  v.  3,  p.  1141;  Sur  la  dissociation  de  Vacide  selenhydrique, 
Paris,  A.  Hermann,  1898. 


200  THERMODYNAMICS  AND  CHEMISTRY. 

The  total  pressure  reduced  to  23°  was  1270  mm.:: 


The  total  pressure  reduced  to  25°  was  1520  mm.g 

X=  0.4200. 
The  total  pressure  reduced  to  22°  was  3016  mm.': 

X=  0.4230. 

TEMPERATURE  475°.    The  total  pressure  of  the  gaseous  mix- 
ture reduced  to  22°  was  1450  mm.  of  mercury  : 

Z=0.3840. 

.  The  total  pressure  reduced  to  23°  was  2556  mm/: 

^=0.3917. 

TEMPERATURE  325°.    The  total  pressure  of  the  gaseous  mix- 
ture reduced  to  17°  was  825  mm.  of  mercury: 


The  total  pressure  reduced  to  15°  was  3240 
X  =  0.206. 

170.  Variation  of  the  solubility  of  a  salt  with  pressure.  —  We 

owe  to  F.  Braun  l  other  interesting  verifications  of  the  principle 
of  the  displacement  of  equilib  ium  by  varying  the  pressure;  they 
are  drawn  from  the  study  of  saturated  solutions. 

Consider  at  a  gi  en  pressure  and  temperature  a  bivariant 
system  whose  two  components  are  a  salt  and  water  and  whose 
two  phases  are  the  solid  salt  and  an  aqueous  solution  ;  when  the 
system  is  in  equilibrium,  the  solution  has  a  definite  concentra- 
tion S;  it  is  saturated  at  the  given  pressure  and  temperature. 

Imagine  that  a  very  small  mass  of  salt  passes  from  the  solid 
phase  into  the  almost  saturated  solution;  the  volume  of  the  solid 
phase  decreases  by  a  quantity  which  is  known  when  the  density 
of  the  salt  is  known  ;  the  volume  of  the  liquid  phase  undergoes  an 

1  F.  BRAUN,  Wiedemann's  Annakn,  v.  30,  p.  250,  1887. 


THE  DISPLACEMENT  OF  EQUILIBRIUM.  201 

increase  which  may  be  calculated  when  we  know  the  law  of  the 
variation  of  the  density  of  the  solution  for  concentrations  near 
to  S;  the  volume  of  the  system  undergoes  a  modification  which 
may  be  either  an  increase  or  a  decrease. 

At  ordinary  temperature  and  pressure,  the  solution  of  alum  or 
of  sodium  sulphate  with  the  molecules  of  water  in  an  aqueous 
solution  almost  saturated  with  the  same  salt  is  accompanied  by  a 
contraction  of  the  system;  in  the  same  conditions,  the  solution  of 
ammonium  chloride  is  accompanied  by  an  expansion. 

At  a  given  temperature  take  a  saturated  solution  of  a  definite 
salt,  in  the  presence  of  an  excess  of  the  salt,  at  the  pressure  TT;  S 
is  the  concentration  of  the  solution.  Give  to  the  pressure  a  value 
TT',  slightly  greater  than  TT,  keeping  the  temperature  constant;  the 
equilibrium  will  be  disturbed  and  the  composition  of  the  solution 
will  vary  until  its  concentiations  has  taken  the  value  S'  which 
corresponds  to  saturation  at  the  pressure  TT'. 

The  modification  undergone  by  the  system  while  the  solution 
changes  from  the  concentration  S  to  the  concentration  S'  should 
be  a  modification  which,  at  constant  pressure  and  temper  lure, 
would  involve  a  decrease  in  volume;  if  the  dissolving  of  the  salt 
in  a  nearly  saturated  solution  takes  place  with  contraction,  this 
modification  consists  in  the  dissohing  of  a  certain  mass  of  salt, 
and  S'  is  greater  than  S;  if  the  dissolving  of  the  salt  in  a  nearly 
saturated  solution  takes  place  with  dilatation,  this  modification 
consists  in  the  precipitation  of  a  certain  quantity  of  salt,  and  S' 
is  less  than  S.  The  following  propositions  may  therefore  be  stated: 

//,  at  a  given  temperature,  the  dissolving  of  a  salt  in  a  nearly- 
saturated  solut'on  is  accompanied  by  contraction,  the  solubility  of 
the  salt  increases  with  pressure;  if,  on  the  contrary,  the  dissolving  of 
the  salt  in  the  nearly  saturated  solution  is  accompanied  by  expansion, 
the  solubility  of  the  salt  decreases  as  the  pressure  increases. 

The  first  case  is  illustrated,  as  we  have  said,  by  alum  and  sodium 
sulphate  with  ten  molecules  of  water;  if  we  compress  very  slowly, 
in  a  piezometer,  a  saturated  solution  of  alum  or  of  sodium  sulphate 
with  ten  molecules  of  water,  in  the  presence  of  an  excess  of  the 
same  salt,  the  solution  will  dissolve  a  new  quantity  of  salt;  it  will 
remain  clear  during  the  compression;  relieved  cautiously  and 
brought  back  to  ordinary  pressure,  it  will  possess  the  properties 


202  THERMODYNAMICS  AND  CHEMISTRY. 

of  a  supersaturated  solution;  the  crystals  still  remaining  in  excess 
will  have  eaten  faces. 

An  illustration  of  the  second  case  is  ammonium  chloride;  if 
a  saturated  solution  of  ammonium  chloride  in  the  presence  of 
crystals  of  this  salt  becomes  supersaturated,  the  solution  will 
deposit  on  the  crystals  a  part  of  the  salt  which  it  contains. 

The  solution  of  sodium  chloride  offers  interesting  peculiarities. 

Let  us  study  at  a  constant  temperature,  as  15°,  the  dissolving 
of  a  small  mass  of  sodium  chloride  in  a  nearly  saturated  solution 
of  this  salt;  this  phenomenon  is  produced  with  contraction  of  the 
system  if  it  takes  place  under  a  constant  pressure  less  than  1530 
atmospheres;  on  the  contrary,  it  is  accompanied  by  dilatation  if 
it  takes  place  under  a  pressure  greater  than  1530  atmospheres; 
when  the  pressure  exceeds  1530  atm.  and  continues  to  increase, 
the  solubility  diminishes;  at  the  constant  temperature  of  15° 
the  pressure  of  1530  atm.  corresponds  to  a  maximum  of  solubility 
of  sodium  chloride  in  water. 

The  existence  of  this  maximum  of  solubility  has  been  shown 
by  F.  Braun;  one  compresses  very  slowly,  to  a  pressure  much 
higher  than  1530  atmospheres,  a  saturated  solution  of  sodium 
chloride  in  the  presence  of  crystals  of  sea-salt;  after  return  to 
the  ordinary  pressure,  the  crystals  placed  in  the  piezometer  are 
examined;  their  faces  are  eaten  and  carry  little  cubical  crystals 
of  sodium  chloride;  the  sodium  chloride  must  therefore  dissolve 
during  a  part  of  the  time  of  compression  and  precipitate  during 
the  rest  of  this  time. 

171.  Displacement  of  equilibrium  by  variation  of  the  tem- 
perature.— The  very  simple  and  most  fruitful  law  of  the  displace- 
ment of  equilibrium  by  the  variation  of  the  pressure  was  stated 
by  H.  Le  Chatelier  l  in  1884;  a  short  time  previously,  J.  H.  Van't 
Hoff  2  had  announced  the  still  more  important  law  of  the  displace- 
ment of  equilibrium  by  variation  of  the  temperature. 

There  are  actually  two  laws  of  the  displacement  of  equilibrium 
by  variation  of  the  temperature ;  the  one  supposes  the  system  kept 
at  constant  pressure,  the  other  at  constant  volume;  these  two 

1  H.  LE  CHATELIER,  Comptes  Rendus,  v.  99,  p.  786,  1884. 

2  J.  H.  VAN'T  HOFF,  Etudes  de  dynamique  chimique,  Amsterdam,  1884. 


THE  DISPLACEMENT  OF  EQUILIBRIUM.  203 

laws  having  exactly  the  same  form,  we  shall  be  content  to  state  the 
first;  it  will  suffice  in  our  statement  to  replace  the  words  constant 
pressure  by  the  words  constant  volume  to  give  the  second  law. 

A  chemical  system  is  in  stable  equilibrium  at  a  given  pressure 
and  at  a  temperature  T;  keeping  the  pressure  constant,  INCREASE 
slightly  the  temperature  to  Tf ;  the  equilibrium  is  disturbed;  in  order 
to  reach  the  new  equilibrium  state  corresponding  to  the  given  pressure 
and  to  the  temperature  Tr ,  the  system  must  undergo  a  certain  change 
of  state;  if  this  change  of  state  was  produced  under  the  given  con- 
stant pressure  and  at  the  invariable  temperature  T',  it  would  be 
accompanied  by  an  ABSORPTION  OF  HEAT. 

172.  Lowering  of  freezing-points  of  solutions. — Let  us  see, 
from  an  example,  the  importance  of  this  law. 

Under  a  given  pressure,  such  as  the  atmospheric  pressure,  and 
at  the  temperature  T,  there  is  stable  equilibrium  in  a  bivariant 
system  formed  by  ice  in  contact  with  a  salt  solution;  s  is  the  con- 
centration of  the  solution. 

Without  changing  the  pressure,  give  the  pressure  a  new  value 
T',  slightly  higher  than  T]  the  equilibrium  is  disturbed;  in  order 
to  reestablish  it,  the  solution  must  assume  a  concentration  sf 
different  from  s. 

Now  the  modification  undergone  by  the  system  while  the 
concentration  of  the  solution  changes  from  s  to  s'  would  absorb 
heat  if  it  took  place  at  constant  temperature  and  pressure;  of  the 
two  changes  of  state  of  which  the  system  is  capable,  fusion  of  a 
part  of  the  ice,  freezing  of  a  part  of  the  solvent,  the  first  only  ful- 
fils the  conditions  which  we  have  indicated;  therefore  the  pas- 
sage of  the  solution  of  concentration  s  to  the  concentration  s'  has 
necessitated  the  fusion  of  a  certain  mass  of  ice,  so  that  s'  is  inferior 
to  s. 

Thus  the  concentration  of  a  salt  solution,  which  can  remain 
tinder  a  given  constant  pressure  in  equilibrium  with  ice,  decreases 
as  the  temperature  increases.  This  may  be  stated  in  the  follow- 
ing way: 

Under  a  given  pressure,  the  freezing-point  of  the  solvent  in  a 
solution  of  given  nature  is  lowered  as  the  solution  becomes  more 
concentrated. 

This  lowering  of  the  freezing-point  of  a  liquid  by  mixing  with 


204 


THERMODYNAMICS  AND  CHEMISTRY. 


it  a  foreign  substance  has  been  known  a  long  time.  Berthollet 
attributes  the  discovery  to  Blagden,  who  in  1788  observed  it  in 
dissolving  salts  in  water.  It  has  since  been  the  object  of  numerous 
investigations.  Let  us  note  in  particular  those  of  Raoult,1  who 
studied  the  freezing  of  solutions  of  water,  benzene,  nitre-benzene, 
ethylene  bromide,  formic  acid,  acetic  acid,  etc.  The  laws  to  which 
these  experiments  led  Raoult  have  become  the  foundation  of  an 
important  branch  of  physical  chemistry,  cryoscopy. 

173.  Lowering  of  the  tension  of  the  saturated  vapor  of  solu- 
tions.— Consider  again  the  preceding  reasonings,  but  replacing  the 
word  ice  by  the  word  vapor;  of  the  two  modifications  which  may 
be  produced  in  the  system,  condensation  of  a  certain  mass  of  vapor, 
vaporization  of  a  certain  quantity  of  solvent,  the  first,  at  constant 
temperature  and  pressure,  liberates  heat,  the  second  absorbs  it; 
therefore  the  passage  of  the  solution  from  concentration  s  to  the 
concentration  s'  has  necessitated  the  vaporization  of  a  part  of  the 
solvent,  so  that  the  concentration  sf  is  greater  than  the  concen- 
tration s.  From  this  is  derived  the  following  proposition: 

Under  a  given  pressure,  the  boiling-point  of  a  solution  of  given 
nature  is  increased  as  the  solution  becomes  more  concentrated. 

If  we  consider  a  solution  of  given  concentration  s,  at  each 
temperature  it  will  have  a  well-defined  pressure  of  saturated 
vapor;  if,  on  the  two  rectangular  coordinate  axes  sOT,  On  (Fig.  53) 

we  lay  off  the  temperatures  T  as 
abscissae  and  the  pressures  n  as  ordi- 
nates,  to  the  concentration  s  there 
will  correspond  a  curve  C  of  ten- 
sions of  the  saturated  vapor;  to  a 
concentration  s'  will  correspond  an- 
other analogous  curve,  C'. 

All   the   curves    C,    C', .  .  .   rise 
from  left  to  right. 

Take  a  particular  value  P  of  the 
pressure  it  and  trace  the  line  PP' 
parallel   to    OT,   for   which   all   the 
points  have  for  ordinate  this  value  ;r=P  of  the  pressure.     This 


1  F.  M.  RAOULT,  Comptes  Rendus,  v.  95  to  99,  1880  to  1884. 


THE  DISPLACEMENT  OF  EQUILIBRIUM.  205 

line  PP'  cuts  the  curve  C,  C", .  .  .  at  the  points  M ,  M ' .  .  .  which 
have  for  respective  abscissae  T,  T', .  .  .  These  temperatures  Tt 
T't .  .  .  are,  under  the  pressure  P,  the  respective  boiling-points 
of  the  solutions  of  concentrations  s,  sf,  .  .  .  According  to  the 
preceding  theorem,  if  s'  is  greater  than  s,  T'  is  greater  than  Tr 
and  the  point  M'  is  to  the  right  of  the  poii.t  M. 

Furthermore,  the  line  TM  certainly  cuts  the  curve  C'  in  a 
point  N  located  below  the  point  M;  but  TM  is  at  the  tempera- 
ture T,  the  tension  of  saturated  vapor  from  the  solution  at  con- 
centration s;  TN  is,  at  the  same  temperature,  the  tension  of  satu- 
rated vapor  from  the  solution  at  concentration  s';  the  following 
theorem  may  therefore  be  stated: 

At  a  given  temperature  the  tension  of  saturated  vapor  from  a 
solution  is  diminished  as  'h  con  entr  tion  of  the  solution  is  increased. 

174.  Dissociation  of  exothermic  compounds  and  formation 
of  endothermic  compounds  by  rise  in  temperature. — Some  purely 
chemical  applications  will  indicate  more  clearly  the  importance 
of  the  law  stated  by  Van'  Hoff. 

Under  constant  pressure,  let  us  study  a  bivariant  or  multi- 
variant  system  in  which  a  certain  chemical  compound  may  be 
formed  or  destroyed;  in  a  given  state,  the  system  encloses  a  mass 
m  of  this  compound;  the  elementary  composition  of  the  system 
is  such  that  the  mass  of  this  compound  would  have  the  value  M 
if  the  combination  were  carried  as  far  as  possible;  let  x  be  the 

,.     m 
ratio  y. 

For  the  system  in  stable  equilibrium  at  the  temperature  T  x 
has  a  value  X;  if  the  temperature  passes  from  the  value  T  to  a 
slightly  higher  value  T',  x  assumes  a  new  value  X'  near  to  X]  is 
X'  greater  o  less  than  X?  Such  is  the  question  which  the  law 
of  displacement  of  equilibrium  by  variation  of  the  temperature 
permits  us  to  answer. 

When  x  passes  from  the  value  X  to  the  value  X',  the  system 
is  the  seat  of  a  certain  reaction;  accomplished  at  constant  pres- 
sure and  temperature,  this  reaction  would  absorb  heat;  therefore 
if  the  formation,  under  constant  pressure,  of  the  compound  con- 
sidered, liberates  heat,  this  reaction  is  a  decomposition  and  X'  is 
less  than  X;  if  the  formation  of  this  compound  under  constant 


206  THERMODYNAMICS  AND  CHEMISTRY. 

pressure  absorbs  heat,  the  reaction  is  a  combination  and  X'  is 
greater  than  X. 

We  have  then  the  double  proposition : 

//,  without  changing  the  pressure,  the  temperature  of  a  system 
containing  an  exothermic  compound  (at  constant  pressure)  is  grad- 
ually raised,  the  proportion  of  the  non-dissociated  compound  is  more 
and  more  diminished. 

If,  without  varying  the  pressure,  the  temperature  of  a  system 
containing  an  endothermic  compound  (at  constant  pressure)  and  the 
elements  whose  combination  may  form  this  compound  is  gradually 
raised,  the  proportion  of  the  compound  in  the  system  is  increased. 

Water  vapor  and  carbonic  acid  are  bodies  which  are  formed, 
under  constant  pressure,  with  liberation  of  heat;  if,  therefore,  at 
a  constant  pressure,  as  atmospheric  pressure,  for  example,  the 
temperature  of  a  system  containing  one  of  these  compounds  is 
raised,  this  compound  will  dissociate  more  and  more  completely, 
as  has  been  verified  by  the  memorable  researches  of  H.  Sainte- 
Claire  Deville. 

175.  Actions  produced  by  a  series  of  electric  sparks;  inter- 
pretation given  by  H.  Sainte-Claire  Deville.  Apparatus  with 
cold  and  with  hot  tubes. — When  a  series  of  electric  sparks  are 
passed  for  a  sufficient  time  through  a  gas  formed  with  liberation 
of  heat,  it  often  happens  that  this  gas  is  more  or  less  completely 
decomposed;  ammonia  gas,  for  example,  is  almost  completely 
decomposed  into  nitrogen  and  hydrogen;  hydrochloric  acid,  on 
the  contrary,  undergoes  only  a  trace  of  decomposition. 

Perrot,1  causing  considerable  quantities  of  water  vapor  to  pass 
between  the  multiple  sparks  of  an  induction  coil,  obtained  a  par- 
tial decomposition  of  water  vapor  into  its  elements.  H.  Sainte- 
Claire  Deville 2  did  not  hesitate  to  see  in  this  experiment  the  ana 
logue  of  Grove's  experiment.  The  spark,  a  line  of  fire  at  very 
high  temperature,  dissociates  the  water  vapor  as  does  the  mass 
of  incandescent  platinum;  the  oxygen  and  hydrogen  liberated 

•PERROT,  Comptes  Rendus,  v.  47,  p.  351,  1858;  Recherches  sur  V action 
chimique  de  Vetincelle  d 'induction  de  I'appareil  de  Ruhmkorff.  Th6se,  Paris, 
1861. 

2  H.  SAINTE-CLAIRE  DEVILLE,  Bibliotheque  universelle,  Archives,  Nouvelle 
p^riode,  v.  6,  p.  267,  1859. 


THE  DISPLACEMENT  OF  EQUILIBRIUM.  207 

are  abruptly  cooled  by  contact  with  the  cool  gases  encountered 
a  few  millimetres  away  from  the  spark;  brought  to  a  temperature 
at  which  their  direct  combination  is  no  longer  produced,  they 
may  elude  observation. 

If  this  explanation  is  correct,  if  the  actions  which  determine 
a  series  of  sparks  are  merely  actions  which  are  produced  at 
a  very  high  temperature  and  which,  thanks  to  the  quickness  of 
cooling,  have  not  the  time  to  be  completely  reversed,  it  should 
be  possible  to  reproduce  these  actions  without  making  any  use 
of  electricity;  for  this  it  would  suffice  to  circulate  the  gases  to  be 
studied  in  a  space  where  a  very  cold  region  is  in  immediate  con- 
tact with  a  very  hot  region. 

Here  is  how  H.  Sainte-Claire  Deville  reali2ed  these  conditions: 

Take  a  porcelain  tube  and  place  it  in  a  furnace,  which  may 
be  heated  to  a  very  high  temperature;  the  ends  of  the  tube  are 
closed  by  corks  each  pierced  with  two  holes.  The  gas  may  be  let 
in  and  out  through  one  hole  of  each  end;  through  the  other  two 
holes  is  passed  a  tube  of  silvered  brass,  8  mm.  in  diameter,  running 
the  length  of  the  porcelain  tube  and  through  which  a  rapid  cur- 
rent of  cold  water  is  passed.  Finally,  two  small  screens  of  un- 
glazed  porcelain  separate  the  cool  portions  of  the  porcelain  tube 
outside  the  furnace  from  the  hot  part  inside. 

"This  brass  tube  even  in  its  hottest  parts  is  cooled  to  about 
10°  by  the  current  of  ccld  water.  The  velocity  of  this  water  is 
such  that  in  passing  through  the  incandescent  tube  the  latter 
does  not  sensibly  heat  it. 

"There  is  thus  in  a  small  space  a  cylindrical  porcelain  surface 
very  strongly  heated  and  a  concentric  brass  surface  very  cold. 

"  .  .  .In  order  to  give  an  idea  of  the  strange  manner  in  which 
this  apparatus  acts,  I  shall  say  that  one  may  with  impunity  cover 
the  metallic  tube  with  the  most  alterable  organic  substances, 
plunge  them  into  a  bright  brazier,  with  which  I  operate,  and  note 
in  this  wray  certain  decompositions.  If  the  layer  of  alterable 
substance  is  sufficiently  thin,  it  will  always  be  protected  against 
the  action  of  the  fire  by  the  current  of  fresh  water  passing  through 
the  metallic  tube.  It  suffices  that  the  latter  has  thin  walls  and 
that  they  be  ?ood  heat  conductors.  The  mass  of  very  hot  gas 
being  quite  insensible  in  comparison  with  the  mass  of  the  cooling 


208  THERMODYNAMICS  AND  CHEMISTRY. 

apparatus,  and  the  conductivity  of  the  gas  being  nearly  zero,  the 
cooling  of  the  matter  experimented  upon  will  always  be  sudden, 
and  one  will  have  quite  the  same  conditions  realized  with  the 
electric  spark." 

176.  Dissociation  of  carbonous  oxide,  of  sulphurous  and  hy- 
drochloric acid  gases.  Synthesis  of  ozone. — If  a  current  of  car- 
bonous acid  gas  is  passed  through  this  apparatus,  the  gases  coming 
from  the  tube  contain  a  certain  quantity  of  carbonic  acid,  while 
the  cold  metallic  tube  becomes  covered  with  a  carbon  deposit; 
the  carbonous  oxide  is  therefore  partially  decomposed  into  car- 
bonic oxide  and  carbon  according  to  the  equation 

2CO  =  CO2+C. 

This  reaction  is  the  same  one  taking  place  when  a  series  of 
electric  sparks  are  passed  through  a  eudiometer  containing  car- 
bonous oxide. 

A  current  of  sulphurous  anhydride  may  be  passed  through  the 
apparatus  with  cold  and  hot  tubes  after  having  plated  the  brass 
tube  with  a  thick  layer  of  pure  silver;  the  silver  has  no  sensible 
action  on  the  sulphurous  acid  at  the  temperature  of  300°  and, 
a  fwti&ri,  at  the  temperature  of  10°  at  which  it  is  kept  during 
these  experiments;  at  th?  end  of  a  certain  time  the  silver  is  found 
to  be  considerably  blackened  by  its  transformation  into  silver 
sulphide  and  covered  with  a  layer  of  sulphuri  anhydride,  which 
absorbs  moisture  readily  from  the  air  and  produces,  in  a  solution 
of  barium  chloride,  an  abundant  precipitrte.  The  sulphurous 
anhydride  has  therefore  been  decomposed  into  sulphuric  anhy- 
dride and  sulphur,  according  to  the  equation 

3SO,=2S03+S. 

By  various  experiments  H.  Sainte-Claire  Deville  showed  that 
this  is  also  the  equation  for  the  partial  decomposition  undergone 
by  sulphurous  anhydride  in  a  eudiometer  in  which  passes  a  series 
of  electric  sparks. 

When  a  train  of  electric  sparks  are  made  to  pass  through  a 
eudiometer  containing  hydrochloric  acid,  a  small  quantity  of  this 
.acid  is  decomposed  into  hydrogen  and  oxygen. 


THE  DISPLACEMENT  OF  EQUILIBRIUM.  209 

This  same  decomposition  takes  place  at  the  hightest  tempera- 
tures which  may  be  produced  by  laboratory  furnaces. 

To  demonstrate  this,  pass  a  current  of  pure,  dry  hydrochloric 
acid  through  the  apparatus  with  cold  and  hot  tubes  after  having 
covered  the  cold  tube  with  a  layer  of  silver  amalgam,  unattackable 
by  the  hydrochloric  acid  at  the  low  temperature  at  which  it  is 
kept.  After  several  hours  the  mercury  and  even  the  silver  have 
some  chloride  on  the  surface,  for  upon  wetting  the  tube  with 
ammonia  the  tube  is  blackened  and  the  ammonia  takes  up  a  small 
amount  of  silver  chloride. 

These  various  experiments  place  beyond  doubt  the  hypothesis 
formulated  by  H.  Sainte-Claire  Deville:  the  eudothermic  decom- 
positions which  are  produced  by  the  passage  of  a  long  series  of 
sparks  within  a  gas  are  due  to  the  high  temperature  produced 
by  the  spark;  they  are  likewise  confirmations  of  the  principle  of 
the  displacement  of  equilibrium  by  variation  of  temperature. 

The  passage  of  a  series  of  electric  sparks  through  a  gaseous 
system  is  not  merely  susceptible  of  producing  certain  decomposi- 
tions; it  may  also  give  rise  to  certain  syntheses.  We  do  not  wish 
to  speak  here  of  sudden  combinations  and  explosives,  such  as  the 
combination  of  oxygen  and  hydrogen  which  a  single  spark  suffices 
to  provoke,  but  slow  combinations  determined  by  the  passage  of 
frequent  electric  sparks,  prolonged  over  several  hours;  the  type 
of  these  syntheses  is  the  partial  transformation  of  oxygen  into 
ozone: 

302=203. 

If  H.  Sainte-Claire  Deville's  way  of  veiwing  this  is  correct, 
these  syntheses  should  not  be  regarded  as  indirect  reactions, 
rendered  possible  by  a  certain  electrical  action,  but  as  reactions 
which  are  produced  directly  at  high  temperatures;  the  apparatus 
with  cold  and  hot  tubes  should  suffice  to  produce  them  without 
making  any  use  of  electricity. 

Troost  and  Hautefeuille  have  in  fact  shown  that  if  a  current 
of  oxygen  is  passed  through  a  hot  tube  brought  to  1300°  or  1400°, 
while  the  cold  tube  is  covered  with  a  layer  of  pure  silver,  there 
collects  on  this  tube,  after  a  certain  time,  silver  dixoide,  certain 
indication  of  a  transformation  of  oxygen  into  ozone  by  contact 
with  the  extremely  hot  porcelain. 


210  THERMODYNAMICS  AND  CHEMISTRY. 

But,  according  to  the  determinations  of  Berthelot,  the  reaction 
302=203 

absorbs  61 .4  cals. ;  hence  the  direct  formation  of  ozone  at  a  higher 
temperature  is  a  remarkable  confirmation  of  the  law  of  the  dis- 
placement of  equilibrium  with  variation  of  temperature. 

177.  Synthesis  of   acetylene. — We  see  from  this  experiment 
that  the  formation  of  ozone  within  the  oxygen  traversed  by  a 
series  of  electric  sparks  should  be  regarded  as  a  reaction  which 
is  produced  of  itself  at  a  high  temperature;   the  same  interpreta- 
tion should  be  accepted  for  a  great  number  of  syntheses  produced 
by  a  series  of  sparks  or  by  the  electric  arc. 

Thus  when  a  current  of  hydrogen  is  passed  between  the  two 
carbon  electrodes  of  an  electric  arc.  acetylene  gas  is  formed,  as 
was  shown  by  Berthelot;1  the  formation  of  acetylene  in  these 
circumstances  should  be  regarded  as  a  reaction  which  produces 
itself  at  the  extremely  high  temperature  of  the  electric  arc. 

According  to  Berthelot,  the  reaction 

2C  +  2H  =  C2H2, 

which  represents  the  formation  of  acetylene,  absorbs  58.1  cals. 
The  formation  of  acetylene  at  the  temperature  of  the  electric  arc 
should  also  be  regarded  as  a  consequence  of  the  law  of  the  dis- 
placement of  equilibrium. 

Analogous  examples  exist  in  great  numbers;  we  shall  limit 
ourselves  to  those  cited  above. 

178.  Case  of  reactions  which   neither    absorb   nor   liberate 
heat. — A  particularly  interesting  case  is  the  one  for  which  the 
compound  contained  in  the  system  is  formed,  under  constant  pressure, 
without  liberation  or  absorption  of  heat;   in   this   case   a  reasoning 
similar  in  all  points  to  that  which  we  have  developed  above  shows 
us  that  X'  can  neither  be  greater  nor  less  than  X;  the  proportion 
of  the  compound  contained  by  the  system,  when  it  is  in  equilibrium 
under  a  given  pressure,  is  independent  of  the  temperature. 

179.  Phenomena  of  etherification. — The  studies  of  Berthelot 
on  etherification  furnish  an  application  of  this  law. 

1  BERTHELOT,  Comptes  Rendus,  v.  54,  pp.  640  and  1042,  1862. 


THE  DISPLACEMENT  OF  EQUILIBRIUM.  211 

The  etherification  of  alcohol  by  acetic  acid  does  not  give  rise 
to  any  appreciable  quantity  of  heat.  If  these  substances  are 
mixed  in  equivalent  proportions  and  left  long  enough  for  equilib- 
rium to  be  established,  it  is  found  that  the  proportions  of  ether- 
ized acid  are  the  following: 

At  room  temperature,  after  16  years 0 . 652 

At  100°,  after  a  very  long  time 0.656 

At  170°,  after  42  hours 0.665 

At  200°,  after  24  hours 0 .673 

At  220°,  after  38  hours 0.665 

These  numbers  should  be  regarded  as  identical. 

180.  Minimum  dissociation  of  hydrogen  selenide. — One  may, 
in  all  the  preceding  statements,  replace  the  words  constant  pres- 
sure by  the  words  constant  volume  without  changing  the  accuracy 
of  these  statements,  so  that  the  following  considerations  are  justi- 
fied: 

At  constant  pressure  raise  the  temperature  of  a  system  which 
contains  liquid  selenium,  hydrogen,  and  gaseous  hydrogen  selenide; 
for  the  system  in  equilibrium  the  ratio  X  between  the  mass  of 
sel  nide  formed  and  he  possible  mass  of  selenide  varies  as  the 
temperature  is  raised;  this  ra  io  increases  at  first,  passes  through 
a  maximum,  then  diminishes  while  the  temperature  continues 
to  rise. 

Ditte  l  was  the  first  to  announce  a  maximum  for  the  ratio  X; 
unfortunately  his  observations  were  erroneous,  due  to  the  partial 
absorption  of  hydrogen  selenide  by  the  liquid  selenium;  H.  Pela- 
bon,2  eliminating  this  source  of  error,  was  able  to  study  the  varia- 
tions of  the  ratio  X  with  the  temperature  and  establish'  d  beyond 
doubt  the  existence  of  a  maximum  for  this  ratio;  this  maximum 
corresponds  to  a  temperature  of  575°,  and  its  value  is  about  0.41. 

It  should  be  concluded  that  there  is  absorption  of  heat  when, 
at  constant  volume  and  at  a  constant  temperature  less  than  575°, 

1  DITTE,  Annales  de  I'Ecok  normale  superieure,  2d  S.,  v.  i,  p.  293,  1873. 

3  H.  PELABON,  Mem.  de  la  Soc.  des  Sciences  physiques  et  naturelles  de  Bor- 
deaux, 5th  S.,  v.  3,  p.  241 ;  Sur  la  dissociation  de  Vacide  selenhydrique,  Paris, 
A.  Hermann,  1898. 


212  THERMODYNAMICS  AND  CHEMISTRY. 

liquid  selenium  and  hydrogen  combine  to  form  hydrogen  selenide; 
on  the  contrary,  when  this  reaction  takes  place  at  a  temperature 
higher  than  575°  it  must  liberate  heat. 

Hautefeuille  had  already  shown  that,  at  constant  pressure  at 
ordinary  temperature,  the  formation  of  hydrogen  selenide  from 
hydrogen  and  liquid  selenium  was  an  endothermic  reaction; 
Fabre  *  has  recently  given  an  exact  determination  of  the  heat  of 
formation  of  hydrogen  selenide  in  these  conditions.  If  it  is  ob- 
served, also,  that  the  combination,  under  constant  pressure,  of 
hydrogen  and  liquid  selenium  is  accompanied  by  almost  no  change 
in  volume,  it  is  seen  that  the  heat  of  formation  under  constant 
pressure  is  sensibly  equal  to  that  under  constant  volume.  Thus 
is  verified  the  first  part  of  the  preceding  statement,  a  consequence 
of  the  principle  of  displacement  of  equilibrium  by  variation  of 
the  temperature. 

181.  Similarity  of  the  preceding  principle  and  Moutier's  Law. 
At  very  low  temperatures  the  principle  of  maximum  work  is 
exact. — This  principle  leads,  for  bivariant  and  multivariant  sys- 
tems, to  conclusions  similar  in  all  respects  to  those  drawn  (Art.  141 
and  142)  for  monovariant  systems  from  Mou tier's  Law;  greatly 
dissociated  at  a  high  temperature,  an  exothermic  compound  re- 
mains less  altered,  in  a  system  in  equilibrium,  as  the  temperature 
is  lowered;  an  endothermic  compound,  on  the  contrary,  is  formed 
in  very  small  quantities  at  low  temperature;  as  the  temperature 
rises  its  stability  increases.  At  a  very  low  temperature,  for  a 
system  in  equilibrium,  one  may  regard  the  dissociation  of  exo- 
thermic compounds  as  almost  null,  the  dissociation  of  endothermic 
compounds  as  almost  complete;  every  endothermic  compound 
is  spontaneously  resolved  into  its  elements;  every  exothermic 
compound  is  formed  spontaneously  at  the  expense  of  its  elements; 
otherwise  expressed,  at  a  very  low  temperature  the  principle  of 
maximum  work  applies  to  all  reactions  without  exception. 

As  the  temperature  is  raised  higher  and  higher,  we  see  increase 
the  number  of  reactions,  decompositions  of  exothermic  compounds, 
or  syntheses  of  endothermic  compounds  which  are  exceptions  to 
the  principle  of  maximum  work.  According  to  the  happy  ex- 

1  FABRE,  Annnles  de  Chimie  et  de  Physique,  6th  S.,^.  10,  p.  482. 


THE  DISPLACEMENT  OF  EQUILIBRIUM.  213 

pression  of  Van't  Hoff,  this  principle  would  be  rigorously  exact  only 
at  0°  absolute. 

However,  if  we  desire  to  understand  fully  the  sense  and  im- 
portance of  this  proposition,  we  must  not  forget  the  existence  of 
states  of  false  equilibrium  of  which  the  preceding  theory  takes 
no  account;  no  reaction  which  contradicts  this  theory  is  ever 
observed,  but,  on  the  other  hand,  a  great  number  of  reactions 
predicted  as  necessary  by  this  theory  do  not  take  place;  the  sys- 
tem which  should  show  them  remains  in  equilibrium. 


CHAPTER  XL 
BIVARIANT  SYSTEMS.     THE  INDIFFERENT  POINT. 

182.  Various  types  of  Invariant  systems :  Solutions  and  double 
mixtures. — A  bivariant  system  is  one  divided  into  a  number  of 
phases  equal  to  the  number  of  independent  components  which 
form  it;    a  system  of  two  independent  components  and  divided 
into  two  phases  is  the  most  generally  studied  type. 

This  type  may  be  divided  into  two  classes. 

It  may  happen  that  one  of  the  two  phases  of  the  system  con- 
sists of  a  definite  compound,  containing  one  of  the  two  components 
or  both,  while  the  other  phase  is  a  mixture  of  the  two  components 
in  varying  proportions;  a  system  which  contains  sodium  chloride 
crystals  in  the  presence  of  an  aqueous  solution  of  sodium  chloride, 
a  system  containing  ice  in  the  presence  of  an  aqueous  solution  of 
potassium  nitrat?,  one  with  crystals  of  hydra  ted  sodium  sulphate 
(Na2SO4-10H20)  in  the  presence  of  an  aqueous  solution  of  sodium 
nitrate  give  us  three  cha  acteri  tic  examples  belonging  to  this 
class  which  we  shall  call  the  class  of  solutions. 

It  may  happen,  on  the  contrary,  that  each  of  the  two  phases 
into  which  the  system  is  divided  is  a  mixture  in  variable  propor- 
tion of  the  two  independent  components;  this  takes  place  when  a 
liquid  mixture  of  water  and  alcohol  is  in  the  presence  of  a  mixed 
vapor  which  contains  these  two  substances  at  once;  this  also  is 
true  when  a  liquid  mixture  of  ether  and  water  divides  into  two 
layers  which  have  different  compositions;  such  systems  form  the 
category  of  double  mixtures. 

183.  Law  of  equilibrium  for  bivariant  systems.     This  equilib- 
brium  is  stable  in  general. — These  two  categories  of  bivariant 

214 


BI VARIANT  SYSTEMS.     THE  INDIFFERENT  POINT.    215 

systems  obey  the  same  law  which  we  have  stated  in  studying  the 
phase  rule  (Art.  96).  If  there  is  given  arbitrarily  a  temperature 
and  pressure,  it  is  possible,  in  general,  to  observe  this  system  in 
equilibrium  at  this  temperature  and  pressure;  the  composition 
of  each  of  the  two  phases  into  which  the  system  in  equilibrium 
is  divided  is  determined  by  the  knowledge  of  this  temperature 
and  pressure. 

In  all  cases,  by  the  words  is  determined  is  not  to  be  understood 
a  determination  which  excludes  all  ambiguity;  it  may  happen, 
and  does  in  certain  cases  which  we  shall  meet  in  this  chapter,  that 
at  a  given  pressure  and  temperature  a  bivariant  system  formed 
of  the  same  independent  components  presents  two  distinct  states 
of  equilibrium  corresponding  to  different  compositions  of  the 
several  phases. 

This  ambiguity  disappears  when  there  is  given  not  only  the 
nature  of  the  two  independent  components,  the  temperature  and 
pressure,  but  also  the  mass  of  each  of  the  independent  compo- 
nents; in  this  case,  not  only  is  the  composition  of  each  of  the 
phases  composing  the  system  in  equilibrium  known,  but  also, 
save  for  a  case  which  we  shall  treat  at  length  in  this  chapter,  the 
mass  of  each  of  the  phases  is  determined. 

Consequently,  when  a  bivariant  system  is  thus  given,  it  is 
impossible,  except  for  the  particular  case  we  have  just  mentioned, 
to  vary  the  masses  of  the  different  phases  which  are  held  in  equilib- 
rium without  causing  their  composition  to  vary,  so  that  at  constant 
pressure  and  temperature  the  system  in  equilibrium  could  not 
undergo  any  modification  without  destroying  the  equilibrium; 
the  exceptional  case  aside,  the  equilibrium  of  a  bivariant  system 
is  not  an  indifferent  equilibrium;  this  distinguishes  the  bivariant 
systems  sharply  from  the  mono  variant  systems. 

It  may  be  shown  that  the  state  of  equilibrium  of  a  bivariant 
system  is  stable  except  for  the  special  case,  when  it  is  indifferent; 
this  proposition  is  of  considerable  importance,  for  it  shows  that 
in  general  one  may  apply  to  bivariant  systems  the  two  laws  of 
displacement  of  equilibrium  by  variation  of  pressure  and  of  the 
displacement  of  equilibrium  by  change  of  temperature;  in  fact, 
in  the  preceding  chapter  we  have  borrowed  several  examples  of 
these  laws  from  the  study  of  bivariant  systems. 


216  THERMODYNAMICS  AND   CHEMISTRY. 

184.  Solutions.  Saturation.  Solubility  curve. — Consider  first 
solutions. 

Two  independent  components,  water,  which  we  shall  denote  by 
the  index  0,  and  an  anhydrous  salt,  which  we  shall  denote  by  the 
index  1,  form  the  system,  which  is  divided  into  two  phases;  the 
one  is  a  solid  salt,  anhydrous  or  hydra  ted,  of  definite  composi- 
tion; the  other  is  a  mixture  of  variable  composition;  this  mixture 
encloses  a  mass  of  water  M0  and  a  mass  of  anhydrous  salt  Mt;  the 

M 

ratio  ^=5  is  the  concentration  of  the  solution. 
M0 

Take  a  pressure  n,  which  we  shall  suppose  always  the  same;: 
it  may  be,  for  example,  atmospheric  pressure;  take  besides  a 
temperature  T;  suppose  that,  at  this  pressure  it  and  temperature 
T,  the  solution  is  in  equilibrium  with  an  excess  of  solid  salt,  case 
in  which  it  is  said  to  be  saturated  with  this  salt;  the  concentra- 
tion of  this  saturated  solution  will  have  a  well-determined  value 
S.  Take  two  rectangular  coordinate 
axes  (Fig.  54);  on  the  axis  of  abscissae 
OT  lay  off  the  values  of  the  tempera- 
ture; on  the  axis  of  ordinates  Os  lay  off 
the  concentrations;  the  concentration  S 
of  the  saturated  solution  at  the  tempera- 
ture T  is  represented  by  a  point  M  having 


T  T    the   coordinates    T,    S;     when,    without 

FIG.  54.  changing  the  pressure,   the  temperature 

T  is  varied,  the  point  M  describes  a  curve  whi  h  is  the  solubility 
curve  of  the  salt  studied  at  the  pressure  considered. 

185.  For  a  hydrated  salt  two  saturated  solutions  correspond 
to  each  temperature.  The  solubility  curve  has  two  branches. — 
What  we  have  just  said  supposes  that  a  single  point  M  corre- 
sponds to  the  temperature  T  or,  in  other  terms,  that  the  concen- 
tration of  the  solution,  saturated  at  the  temperature  T,  has  a  value 
determined  without  ambiguity.  If  the  solid  precipitate  enclosed 
by  the  system  is  an  anhydrous  salt,  the  above  is  certainly  true; 
but  it  may  be  otherwise  if  this  precipitate  is  a  hydrated  salt;  it 
may  happen,  in  this  case,  that  to  the  same  temperature  T  correspond 
two  distinct  saturated  solutions,  one  of  concentration  Slt  the  other  of 


BIVARIANT  SYSTEMS.     THE  INDIFFERENT  POINT.    217 


° 


FlG- 


greater  concentration,  S2;  the  first  richer  in  water  than  the  hydrated 
salt,  the  other  less  rich  in  water  than  the  hydrated  salt. 

These  two  solutions  are  represented  by  the  two  points  Ml 
and  M2  (Fig.  55)  which  have  the  same  abscissa 
T  and  the  ordinates  Sl  and  S2;  when  the  tem- 
perature T  varies,  these  two  points  M1}  M2 
describe  two  curves  Cv  C2,  which  together 
compose  the  solubility  curve  for  the  hydrate; 
the  lower  branch,  Cv  represents  the  saturated 
solutions  richer  in  water  than  the  hydrate; 
the  upper  branch,  C2,  represents  the  saturated 
solutions  less  rich  in  water  than  the  hydrate. 
It  may  be  said  that  the  lower  branch  alone 
exists  for  the  case  in  which  the  precipitate  is 
an  anhydride  salt;  it  exists  also  alone  for  a 
great  number  of  cases  in  which  the  precipitate  is  an  hydrated 
salt. 

1 86.  Non-saturated  and  supersaturated  solutions.— The  vari- 
ous equilibrium  states  of  which  we  have  just  spoken  are  all  stable. 
If  a  small  quantity  of  solid  salt  precipitates  from  a  saturated 
solution,  the  solution  is  brought  in  o  a  state  in  which  it  can  no 
longer  abandon  solid  salt,  but  where  it  dissolves  that  which  is 
thrown  into  it,  a  fact  we  express  by  saying  it  is  unsaturated.  If 
a  small  quantity  of  solid  salt  is  dissolved  in  a  satuiated  solu- 
tion, the  solution  is  brought  at  once  into  a  state  where  it  is  im- 
possible for  it  to  dissolve  the  least  solid  particle;  according  to  the 
predictions  of  thermodynamics,  it  ought  to  give  up  the  salt  it 
contains  in  excess  and  return  to  the  concentration  corresponding 
to  saturation;  we  know  that  this  modification  is  not  always  pro- 
duced, and  that  the  solution  may  remain  in  the  state  of  false  equi- 
librium; it  is  then  said  to  be  supersaturated. 

A  saturated  solution  therefore  becomes  non-saturated  by  the 
subtraction  of  a  small  quantity  of  solid  salt,  and  supersaturated 
by  the  addition  of  a  small  quantity  of  the  same  salt. 

If  a  solution  is  richer  in  water  than  the  solid  salt  precipitated, 
which  is  always  the  case  for  anhydrous  salts,  the  addition  of  a 
small  quantity  of  salt  to  the  solution  increases  its  concentration; 
if,  on  the  contrary,  the  solution  is  less  rich  in  water  than  the  solid 


218 


THERMODYNAMICS  AND  CHEMISTRY. 


salt,  the  addition  of  a  small  quantity  of  this  salt  to  the  solution 
decreases  its  concentration. 

At  this  point  we  may  evidently  state  the  following  proposi- 
tions : 

If  a  saturated  solution  is  represented  by  a  point  of  the  lower 
branch  C±  of  the  solubility  curve,  this  solution  becomes  non- 
saturated  when  the  concentration  is  diminished  and  supersaturated 
when  the  concentration  is  increased;  if,  on  the  contrary,  a  satu- 
rated solution  is  represented  by  a  point  on  the  upper  branch  C2 
of  the  solubility  curve,  this  solution  becomes  non-saturated  when 
the  concentration  is  increased  and  supersaturated  when  the  con- 
centration is  diminished. 

Otherwise  expressed,  the  unsaturated  solutions  are  represented 
by  the  points  of  the  plane  TOs  (Fig.  56)  which  are  situated  below  the 
lower  branch  Ct  or  above  the  upper  branch  C2  of  the  solubility  curve; 
the  supersaturated  solutions  are  represented  by  the  points  situated 
between  the  two  branches. 


Super-saturated 


FIG.  56. 


Super-saturated 


TO  T 

FIG.  57. 


For  the  case  in  which  only  the  lower  branch  exists,  as  for  the  satu- 
rated solutions  of  anhydride  salts  and  for  the  saturated  solutions 
of  a  great  number  of  hydra  ted  salts,  the  non-saturated  solutions 
are  represented  by  the  points  of  the  plane  TOs  (Fig.  57)  which  are 


BIVARIANT  SYSTEMS.    THE  INDIFFERENT  POINT.     219 

above  the  solubility  curve  C,  and  the  supersaturated  solutions  by  the 
points  which  are  below  this  curve. 

187.  Heat  of  solution  in  saturated  solutions. — When  a  very 
small  mass  of  salt  m  passes,  at  the  temperature  T,  from  the  pre- 
cipitate state  into  a  solution  nearly  saturated  at  this  temperature 
T,  the  phenomenon  is  accompanied  by  a  certain  absorption  of 
heat;    the  quantity  of  heat  absorbed  which,  other  things  being 
equal,  is  proportional  to  the  small  mass  m,  depends  on  the  tem- 
perature T  at  which  the  phenomenon  is  produced;  this  quantity 
of  heat  absorbed  may  be  represented  by  the  product  mL,  L  being 
a  fixed  coefficient  for  a  definite  temperature,  but  variable  with  the 
temperature;   L  is  what  is  called  the  heat  of  solution  in  saturated 
solution  of  the  salt  considered,  at  the  temperature  T. 

The  quantity  of  heat  absorbed  of  which  we  have  just  spoken 
is,  in  certain  cases,  negative  in  other  words,  the  dissolving  of  a 
salt  in  an  almost  saturated  solution  may  be  accompanied  by  liber- 
ation of  heat;  in  this  case  the  heat  of  solution  L  is  negative. 

It  is  evident  that  if,  at  a  definite  temperature,  there  exist 
two  distinct  saturated  solutions  of  concentrations  Slt  S2,  there 
correspond  to  these  two  solutions  two  distinct  heats  of  solution 
Lt,  I*. 

188.  Displacement  of  equilibrium  by  variation  of  temperature. 
— The  equilibrium  states  which  we  have  just  studied  being  all 
stable,  we  may  apply  to  them  the  law  of  displacement  of  equi- 
librium by  variation  of  the  temperature. 

A  system,  containing  the  salt  spoken  of  in  contact  with  the 
solution,  is  in  equilibrium  at  the  temperature  T;  the  saturated 
solution  has  the  concentration  S;  without  changing  the  pressure, 
bring  the  temperature  to  a  value  T'  a  little  higher  than  T\  the 
equilibrium  is  broken  and  there  is  produced  in  the  system  a 
change  of  state  which  brings  the  concentration  to  the  value  Sf, 
characterizing  the  saturated  solution  at  the  new  temperature  T'. 

If  this  same  change  of  state  is  produced  without  variation  of 
temperature,  it  should  absorb  heat;  this  change  of  state  consists 
therefore  in  the  solution  of  a  small  quantity  of  salt  if  the  heat  of 
solution  in  saturated  solution  is  positive;  it  consists  in  the  pre- 
cipitation of  a  small  quantity  of  salt  if  the  heat  of  solution  in 
saturated  solution  is  negative. 


220  THERMODYNAMICS  AND  CHEMISTRY. 

We  recall  that  the  mixture  of  a  small  quantity  of  precipitate 
with  the  solution  increases  the  concentration  of  this  solution  if  it 
is  richer  in  water  than  the  precipitate,  and  decreases  the  concen- 
tration of  the  solution  if  it  is  less  rich  in  water  than  the  precipitate; 
we  may  state  the  following  propositions: 

//  the  heat  of  solution  in  saturated  solution  is  positive,  the  lower 
branch  of  the  solubility  curve  rises  from  left  to  right,  the  upper  branch 
of  the  solubility  curve  descends  from  left  to  right.  If  the  heat  of 
solution  in  saturated  solution  is  negative,  the  lower  branch  of  the  solu- 
bility curve  descends  from  left  to  right;  the  upper  branch  of  the  solu- 
bility curve  rises  from  left  to  right. 

Let  us  make  some  applications  of  this  proposition  to  the  lower 
branch  of  the  solubility  curve,  the  only  one  existing  for  the  case 
in  which  the  precipitate  is  anhydrous,  and  for  a  great  number  of 
cases  where  it  is  hydrated. 

Most  salts  dissolve  in  water  with  absorption  of  heat;  also  most 
solubility  curves  rise  from  left  to  right;  the  salt  is  more  soluble 
as  the  temperature  is  higher. 

Sulphate  of  sodium  gives  us  an  example,  however,  of  a  salt 
less  soluble  with  rise  of  temperature. 

At  temperatures  below  23°  a  solution  of  sodium  sulphate  re- 
mains in  equilibrium  in  contact  with  a  precipitate  of  hydrated 
sodium  sulphate,  Na2SO4  •  10H2O ;  this  salt  dissolves  with  absorp- 
tion of  heat;  its  solubility  increases  when  the  temperature  rises. 
At  temperatures  higher  than  23°  sodium  sulphate  with  ten  mole- 
cules of  water  cannot  be  absorbed  in  equilibrium  in  contact  with 
a  solution  of  sodium  sulphate;  on  the  other  hand,  the  latter  may 
rest  in  equilibrium  in  contact  with  a  precipitate  of  anhydrous 
sodium  sulphate;  the  solubility  of  anhydrous  sodium  sulphate 
diminishes  with  increase  of  temperature;  the  heat  of  solution  of 
anhydrous  sodium  sulphate  is  negative,  as  has  been  shown  by 
Pauchon.1 

Calcium  hydrate  and  cerium  sulphate  behave  like  anhydrous 
sodium  sulphate. 

Calcic  orthobutyrate  with  one  molecule  of  water  has  a  solubility 
which  decreases  as  the  temperature  rises  to  60°;  at  60°  this  solu- 

1  PAUCHON,  Comytes  Rendus,  v.  97,  p.  1555,  1883. 


BIVARIANT  SYSTEMS.     THE  INDIFFERENT  POINT.    221 

bility  passes  through  a  minimum;  it  then  increases  with  the  tem- 
perature; the  law  of  the  displacement  of  equilibrium  by  variation 
of  the  temperature  gives  us  then  the  following  information: 

Above  60°  calcic  orthobutyrate  dissolves,  in  nearly  saturated 
solution,  with  liberation  of  heat;  at  60°  the  heat  of  solution  in 
saturated  solution  is  equal  to  0;  beyond  60°  this  heat  becomes 
positive. 

Chancel  and  Parmentier l  have  verified  experimentally  the 
first  part  of  this  statement. 

189.  Precautions  regarding  the  use  of  the   preceding  law. — 
The  law  of  the  displacement  of  equilibrium  by  variation  of  the 
temperature  is  an  exact  theorem,  which  leads  surely  to  just  con- 
clusions, provided  that,  in  applying  it,  the  exact  conditions  indi- 
cated in  its  statement  are  fulfilled;   without  this  precaution  one 
may,  by  an  unjustified  application  of  this  principle,  draw  false 
conclusions,  as  for  example: 

Calcic  isobutyrate  with  five  molecules  of  water  is  the  more 
soluble  the  higher  the  temperature  is  raised;  consequently  the  heat 
of  solution  of  this  salt  in  saturated  solution  is  positive;  Chancel 
and  Parmentier,2  having  measured  the  heat  of  solution  of  calcic 
isobutyrate,  found  it  negative  and  concluded,  therefore,  that  the 
law  of  displacement  of  equilibrium  by  variation  in  temperature 
was  not  always  exact;  Le  Chatelier  3  pointed  out  very  justly  that 
these  physicists  had  measured  not  the  heat  of  solution  in  satu- 
rated solution,  but  the  heat  of  solution  in  very  dilute  solution,  a 
quantity  which  may  be  very  different  from  the  first,  which  may 
even  have  another  sign;  by  direct  experiment  he  proved  that 
the  heat  of  solution  of  hydrated  calcium  isobutyrate,  in  saturated 
solution,  is  positive  as  required  by  the  law  of  the  displacement  of 
equilibrium  by  variation  of  the  temperature. 

190.  The  two  branches  of  the  solubility  curve  of  a  hydrate 
join  each  other  at  the  indifferent  point  where  the  saturated  solu- 
tion has  the  same  composition  as  the  hydrate. — Let  us  take  a 


1  CHANCEL  and  PARMENTIER,  Comptes  Rendus,  v.  104,  pp.  474  and  881» 
1887. 

2  Ibid. 

3  H.  LE  CHATELIER,  Comptes  Rendus,  v.  104,  p.  679,  1887. 


222 


THERMODYNAMICS  AND   CHEMISTRY. 


S2 


T 
FIG.  58. 


hydra  ted  salt  whose   solubility  curve   consists  of  two  branches; 

suppose  that  the  heat  of  solution  in 
saturated  solution  be  positive  for 
both  branches.  The  lower  branch, 
Cx  (Fig.  58),  rises  from  left  to  right; 
the  upper  branch,  C2}  descends  from 
left  to  right. 

To  the  same  temperature  T  cor- 
respond a  point  Ml}  of  ordinate  Slt 
on  the  branch  C1}  and  a  point  M2,  of 
ordinate  S2J  on  the  branch  C2;  as 
the  temperature  rises,  the  two  points 
T  Mlt  M2  approach  each  other  and  the 
two  concentrations  S1}  S2  approach 
each  other. 

May  it  happen  that  at  some  temperature  6  the  two  points 
Mlf  M2  unite  in  a  single  point  7,  that  the  two  concentrations  Slf 
S2  assume  a  common  value  27 

The  concentration  St  is  the  concentration  of  a  solution  richer 
in  water  than  the  hydra  ted  salt  with  which  it  is  saturated;  the 
concentration  S2  is  the  concentration  of  a  solution  less  rich  in 
water  than  the  same  hydra  ted  salt;  if  these  two  concentrations 
Sif  S2  approach  a  common  limit  2  ',  2  is  certainly  the  concentration 
of  a  solution  having  exactly  the  same  composition  as  the  hydrated 
salt  with  which  it  is  saturated. 

Thus  the  two  branches  of  the  solubility  curve  of  a  hydrate  may, 
for  a  certain  value  6  of  the  temperature,  unite  in  a  common  point  I, 
a  point  where  the  saturated  solution  has  the  same  composition  as  the 
hydrate  in  contact  with  which  it  remains  in  equilibrium. 

In  what  manner  is  the  junction  of  the  two  branches  of  the 
solubility  curve  made?  One  might  be  tempted,  in  replying  to 
this  question,  to  apply  again  to  each  of  these  two  branches  the 
law  of  the  displacement  of  equilibrium  with  change  of  temperature; 
this  would  be  an  unwarranted  application  of  this  principle  ;  in  fact, 
the  equilibrium  state  of  the  saturated  solution  at  the  temperature 
0  is  no  longer  a  state  of  stable  equilibrium;  the  saturated  solution 
having,  at  this  temperature,  the  same  composition  as  the  precipi- 
tate, one  may,  without  varying  the  composition  of  the  two  phases 


B1VARIANT  SYSTEMS.    THE  INDIFFERENT  POINT.    223 

and,  moreover,  not  disturbing  the  equilibrium,  suppose  that  a  cer- 
tain mass  of  hydrated  salt  dissolves  or  precipitates;  it  is  there- 
fore clear  that  the  solution  saturated  at  0  degrees  is  in  indifferent 
equilibrium  with  the  hydrated  solid  salt;  also,  we  shall  give  the 
name  indifferent  paint  to  the  point  /,  of  coordinates  6,  I,  which 
represents  this  solution. 

The  law  of  the  displacement  of  equilibrium  by  variation  of 
temperature  being  unable  to  instruct  us  on  the  behavior  of  the 
solubility  curve  in  the  neighborhood  of  the  point  7,  we  should  seek 
this  information  from  a  special  theorem;  this  special  theorem  has 
been  indicated  by  J.  Willard  Gibbs,  and  teaches  us  that — 

The  two  branches  Cit  C2  of  the  solubility  curve  of  the  hydrate  join 
each  other  at  the  point  I,  so  as  to  form  a  curve  without  cusp,  giving 
at  the  point  I  a  tangent  parallel  to  Os. 

igi.  The  temperature  of  junction  is  the  aqueous  fusing-point 
of  the  hydrate.— The  solubility  curve  CJC2  divides  the  plane  into 
two  regions;  one  of  these  regions,  cross-hatched  in  Fig.  59,  is  in 


the  concavity  of  this  curve;  every  point  in  this  region  represents 
a  supersaturated  solution  of  the  hydrate,  the  other  region  repre- 
sents, by  its  various  points,  all  the  non-saturated  solutions  of  the 
hydrate. 

Parallel  to  OT  trace  the  line  22' ,  whose  various  points  have 
for  constant  ordinate  the  concentration  of  a  solution  of  the  same 
composition  as  the  hydrate;  this  line  passes  through  the  point  /; 


224  THERMODYNAMICS  AND   CHEMISTRY. 

the  points  in  this  line  having  abscissae  less  than  6  are  to  the  left 
of  7  and  represent  supersaturated  solutions;  the  points  having 
abscissae  greater  than  6,  to  the  right  of  the  point  7,  represent  non- 
saturated  solutions. 

Take  a  solution,  of  concentration  I,  separated  from  any  solid 
precipitate;  at  a  temperature  higher  than  6  this  solution  will  be 
in  equilibrium;  but  if  the  temperature  falls  below  6,  this  solution, 
supersaturated,  can  no  longer  remain  in  equilibrium,  except  for 
a  phenomenon  of  false  equilibrium;  it  may  precipitate  the  hydrate, 
and  as  this  precipitation  does  not  alter  its  composition,  the  modi- 
fication will  continue  until  all  the  liquid  is  used;  6  is  therefore  the 
temperature  at  which  a  solution  of  the  same  composition  as  the  hy- 
drate solidifies. 

Take  also  a  certain  mass  of  hydrate  in  the  solid  state  and  free 
from  all  trace  of  solution ;  at  a  temperature  less  than  6  this  hydrate 
cannot  undergo  aqueous  fusion,  for  the  solution  engendered,  hav- 
ing a  concentration  21,  would  be  supersaturated  and  would  resolidify ; 
on  the  contrary,  at  a  temperature  higher  than  6,  if  this  hydrate 
could  be  observed  in  equilibrium,  this  state  of  equilibrium  would 
be  unstable;  if  the  hydrate  undergoes  a  trace  of  aqueous  fusion, 
the  solution  resulting,  of  concentration  I,  would  be  non-saturated; 
it  would  commence  to  dissolve  a  new  mass  of  hydrate;  this  dis- 
solving not  changing  the  composition  of  the  solution,  the  dissolving 
will  continue  until  all  the  hydrate  is  fused;  the  temperature  6  is 
therefore  the  temperature  at  which  the  solid  hydrate  undergoes  com- 
plete aqueous  fusion. 

192.  Experimental  investigations  of  Guthrie,  Roozboom,  and 
other  observers. — The  ideas  which  we  have  just  set  forth  exist 
in  germ  in  the  theoretical  works  of  J.  Willard  Gibbs,  but  they  have 
been  brought  to  light  largely  by  the  theoretical  and  experimental 
researches  of  Roozboom  and  Guthrie. 

In  1884  Guthrie  *  described  the  indifferent  point  of  ethylamine 
hydrate,  indifferent  point  which  corresponds  to  the  temperature 
—  8°;  in  1885  Roozboom  2  studied  the  indifferent  points  of  hydro- 
chloric and  hydrobromic  hydrates;  in  1889,  in  a  work  of  capital 

1  GUTHRIE,  Philosophical  Magazine,  5th  S.,  v.  18,  p.  22,  1884. 
8  ROOZBOOM,  Recueil  des  Travaux  chimiqu.es  des  Pays  Bas,  v.  3,  p.  84,  1884; 
V.  4,  p.  102,  1885. 


BIVARIANT  SYSTEMS.     THE  INDIFFERENT  POINT.    225 

importance,1  he  found  -f30°.2  C.  the  temperature  of  the  indifferent 
point  of  the  hydrate  CaCl2-6H2O. 

Pickering2  recognized  for  the  sulphuric  hydrates  SO3-5H3O 
and  SO3-2H2O  the  two  branches  Clt  C2  of  the  solubility  curve, 
and  he  could  follow  each  of  these  branches  over  a  considerable 
temperature  range;  for  the  hydrate  SO3-H2O  he  found  an  indi- 
cation of  the  existence  of  the  upper  branch  for  solutions  more 
concentrated  than  the  hydrate. 

Pickering  3  likewise  undertook  the  study  of  the  combinations 
which  the  amines  form  with  water,  a  subject  which  had  already 
furnished  Guthrie  examples  of  indifferent  points;  Pickering  found 
in  his  turn  the  existence  of  these  points. 

In  a  very  important  investigation  on  the  hydrates  of  ferric 
chloride,  Roozboom  4  showed  the  existence  of  an  indifferent  point 
for  each  of  the  four  hydrates  which  ferric  chloride  may  form. 
These  indifferent  points  correspond  to  the  following  temperatures: 

For  Fe2Cl6-12H2O  6  =+37°    C.  approx. 

Fe2Cl6-  7H2O  0=+32.5 

Fe2Cl6.   5H2O  0=+56 

Fe2CV  4H2O  0=+73.5 

Van't  Hoff  and  Meyerhoffer  5  recognized  the  existence  of  the 
two  branches  of  the  solubility  curve  and  of  the  indifferent  point 
for  the  hydrate  MgCl-12H2O;  this  indifferent  point  corresponds 
to  the  temperature  -16°.3  C. 

Finally,  Le  Chatelier  6  has  studied  with  great  care  the  solu- 
bility of  lithium  borate  in  water;  lithium  borate  furnishes  the 
hydrate  Li2B2O4-16H2O;  the  solubility  curve  of  this  hydrate  is 
composed  of  two  branches;  the  lower  branch,  C1;  corresponding 

1  ROOZBOOM,  Recueil  des  Travaux  chimiques  des  Pays  Bas,  \.  8,  p.  1,  1889 ; 
Archives  neerlandaises  des  Sciences  exactes  et  naturelles,  v.  23,  p.  199,  1889; 
Zeitschrift  fur  physikalische  Chemie,  v.  4,  p.  31,  1889. 

2  PICKERING,  Journal  of  Chemical  Society,  v.  57,  p.  338,  1890. 

3  Ibid.,  v.  63,  pp.  141  and  890,  1893. 

4  ROOZBOOM,   Archives  neerlandaises,   etc.,   v.    28,    1892;    Zeitschrift  filr 
physikalische  Chemie,  v.  10,  p.  447,  1892. 

8  VAN'T  HOFF  and  MEYERHOFFER,  Sitzungsberichte  der  Berliner  Akad., 
Feb.  4  and  18,  1897. 

8  H.  LE  CHATELIER,  Comptes  Rendus,  v.  124,  p.  1091,  1897. 


£26  THERMODYNAMICS  AND  CHEMISTRY. 

to  solutions  less  concentrated  than  the  hydrate,  could  be  fol- 
lowed from  the  temperature  —60°  C. ;  the  upper  branch,  C2, 
corresponding  to  solutions  more  concentrated  than  the  hydrate, 
could  be  followed  from  a  point  whose  abscissa  corresponds  to  the 
temperature  +34°  C.;  these  two  curves  unite  in  an  indifferent 
point  7,  whose  abscissa  corresponds  to  the  temperature  +47°  C.; 
the  behavior  of  the  two  curves  in  the  neighborhood  of  the  point  / 
indicates  clearly  that  they  meet  at  this  point  and  that  their  com- 
mon tangent  is  parallel  to  Os. 

The  hydrates  are  not  the  only  substances  which  give  us  such 
phenomena;  every  time  that  one  may  dissolve  in  variable  pro- 
portion in  a  liquid  0  a  substance  1  susceptible  of  forming  with 
this  liquid  a  solid  compound  2  of  definite  composition,  one  may 
repeat  for  these  three  substances  0,  1,  2  all  that  we  have  just  said 
about  water,  an  anhydrous  salt,  and  the  hydrate  formed  by  their 
union. 

Iodine,  dissolved  in  liquid  chlorine,  may  give  iodine  chloride, 
Id,  capable  of  being  deposited  in  the  solid  state;  this  solid  chloride 
may  exist  in  two  allotropic  forms  denoted  by  the  symbols  ICla 
and  IC1/?;  the  first  form  has  the  fusing-point  +  27°.2  C.,  and  the 
second  the  fusing-point  +13°.9  C.;  Stortenbeker  1  has  shown  that 
each  of  these  two  temperatures  corresponds  to  an  indifferent  point, 
the  one  for  the  solubility  curve  of  ICla  in  liquid  chlorine,  the  one 
for  the  solubility  curve  of  ICl^  in  the  same  solvent. 

The  substance  0  may  be  a  melted  anhydrous  salt,  the  sub- 
stance 1  another  anhydrous  salt,  the  substance  2  a  double  salt 
formed  by  the  combination  of  the  first  two  in  definite  proportions; 
Le  Chatelier  2  has  studied  several  systems  of  this  kind. 

The  solution  of  lithium  carbonate  in  melted  potassium  carbo- 
nate gives  a  solid  double  salt  whose  formula  is  KLiCO2;  the  tem- 
perature of  the  indifferent  point  is  515°  C.  The  melted  mixture 
of  sodium  borate  and  sodium  pyrophosphate  gives  a  double  salt 
formed  by  the  union  of  one  molecule  of  each  of  the  simple 
salts;  the  temperature  of  the  indifferent  point  is  about  960° 
C.  Besides  these  examples  furnished  by  melted  salts,  we  may 

1  W.  STORTENBEKER,  Recueil  des  Travaux  chimiques  des  Pays  Bas,  v.  6, 
1888;  Zeitschrift  fur  physikalische  Chemie,  v.  3,  p.  11,  1888. 

2  H.  LE  CHATELIER,  Comptes  Rendus,  v.  118,  p.  801,  1894. 


BIVARIANT  SYSTEMS.     THE  INDIFFERENT  POINT.    227 

note  the  case  studied  by  Kuriloff,1  the  mixture  of  picric  acid, 
C6H2-(NO2)3-OH,  and  /?-naphtol,  C10H7OH,  giving  the  substance 
C6H2(NO2)3OHC10H7OH,  which  in  the  presence  of  a  liquid  mixture 
of  picric  acid  and  /?-naphtol  has  an  indifferent  point  very  sharply 
marked  at  the  temperature  +157°  C. 

193.  Indifferent  point  of  a  double  mixture.— A  solution  sup- 
porting the  pressure  TT  and  brought  to  the  temperature  T  is  in 
indifferent  equilibrium  in  contact  with  a  hydra  ted  salt  if,  at  this 
temperature  and  under  this  pressure,  the  saturated  solution  has 
the  same  composition  as  the  hydrate;   when  a  double  mixture  is 
in  equilibrium  at  the  pressure  n  and  temperature  T  the  compo- 
sition of  each  of  the  two  phases  into  which  it  is  divided  is  deter- 
mined; if  these  two  phases  have  the  same  composition,  this  state 
of  equilibrium  is  indifferent. 

Suppose,  for  instance,  that  a  mixture  of  volatile  liquids  is  in 
the  presence  of  the  mixed  vapor  which  it  emits;  at  the  tempera- 
ture T  and  pressure  TT  the  liquid  mixture  and  the  mixed  vapor 
which  remain  in  equilibrium  have  definite  compositions;  if  the 
liquid  mixture  and  the  mixed  vapor  happen  to  have  the  same 
composition  at  a  certain  temperature  and  pressure,  the  equilib- 
rium of  the  system  for  this  temperature  and  pressure  is  evidently 
indifferent;  it  is  in  fact  clear  that,  without  changing  the  compo- 
sition of  any  of  the  phases,  consequently  without  disturbing  the 
equilibrium  of  the  system,  we  may  either  vaporize  a  part  of  the 
liquid  mixture  or  condense  a  part  of  the  mixed  vapor. 

194.  Two  theorems  of  Gibbs  and  of  Konovalow. — Under  what 
circumstances  shall  we  observe  such  a  state  of  indifferent  equi- 
librium?   Two    important   theorems,    discovered   by    J.    Willard 
Gibbs,  found  anew  by  D.  Konovalow,  give  us  this  information. 
Here  are  these  two  theorems: 

FIRST  THEOREM  OF  GIBBS  AND  KONOVALOW. — Under  a  con- 
stant pressure  cause  the  composition  of  the  liquid  mixture  to  vary 
in  a  well-defined  way;  the  boiling-point  of  this  mixture  changes',  if, 
for  a  certain  composition  of  the  liquid  mixture,  the  boiling-point 
passes  through  a  maximum  or  minimum,  this  liquid  mixture  gives 
off  a  saturated  vapor  of  the  same  composition,  and  reciprocally. 

1  KURILOFF,  Zettschrift  /.  physikalische  Chemie,  v.  23,  pp.  90  and  673, 
1897. 


228 


THERMODYNAMICS  AND  CHEMISTRY. 


SECOND  THEOREM  OF  GIBBS  AND  KONOVALOW. — At  constant 
temperature  cause  the  composition  of  the  liquid  mixture  to  vary  in  a 
definite  way;  if,  for  a  certain  composition  of  the  liquid  mixture,  the 
tension  of  the  saturated  vapor  passes  through  a  maximum  or  a  mini- 
mum, this  liquid  mixture  gives  off  a  saturated  vapor  of  the  same 
composition,  and  reciprocally. 

195.  Application  of  the  first  theorem  to  mixtures  of  volatile 
liquids. — Let  us  consider  in  detail  the  consequences  of  these  two 
important  theorems  beginning  with  the  first. 

Take  a  liquid  mixture  containing  two  substances  1  and  2;  a 
gramme  of  this  mixture  contains  X  grammes  of  the  substance  2 
and  (l—X)  grammes  of  substance  1;  in  just  the  proportion  that 
the  substance  2  increases  in  the  mixture,  X  will  increase;  starting 
from  the  value  0  at  the  instant  the  liquid  contains  only  the  sub- 
stance 1  in  a  state  of  purity,  X  approaches  1  as  the  mixture  ap- 
proaches the  substance  2  taken  in  the  state  of  purity. 

Consider  this  liquid  mixture  at  the  constant  pressure  TT;  for 
each  value  of  X  will  correspond  a  boiling-point  T;  if  we  take  X 

(Fig.  60)  for  abscissa  and  T  for  ordi- 
nate  of  a  certain  point  M,  the  locus 
of  the  points  M  will  be  a  curve  C; 
this  curve  will  start  from  the  point  Mlt 
which  has  for  abscissa  0  and  for  ordi- 
nate the  temperature  T,  the  boiling- 
point,  under  the  constant  pressure  TT; 
of  the  liquid  1  taken  in  the  state  of 
purity;  it  ends  at  the  point  M2,  which 
has  1  for  abscissa  and  for  ordinate  the 
temperature  T2,  boiling-point,  at  the 
same  pressure  TT,  of  the  liquid  2  taken 
in  the  state  of  purity. 
At  the  temperature  T  and  under  the  pressure  TT  a  gramme  of  the 
saturated  vapor  in  equilibrium  with  the  liquid  of  concentration 
X  contains  x  grammes  of  the  substance  2  and  (1— 2)  grammes  of 
substance  1;  take  a  point  m,  in  the  plane  XOT,  having  x  as  ab- 
scissa and  T  for  ordinate;  this  point  m  will  correspond  to  the  point 
M;  the  ensemble  of  two  corresponding  points  M,  m,  having  a  com- 
mon abscissa,  will  inform  us  as  to  the  composition  of  the  liquid 


M, 


B1VARIANT  SYSTEMS.     THE  INDIFFERENT  POINT.    229 

mixture  and  of  the  mixed  vapor  which  may  coexist  in  equilibrium 
under  the  pressure  TT  at  the  temperature  T.  While  the  point  M 
describes  the  curve  C  from  M±  to  M2,  the  point  m  describes  an- 
other curve,  c,  which  likewise  joins  the  point  M^  to  the  point  M2. 

Let  us  suppose,  to  speak  definitely,  that  the  substance  2  is, 
under  the  pressure  TT,  less  volatile  than  the  substance  1 ;  its  boiling- 
point  T2  at  this  pressure  will  be  higher  than  the  boiling-point  Tl 
of  the  liquid  1.  Three  special  cases  may  then  arise,  concerning 
which  the  principles  of  thermodynamics  give  us  the  following 
information : 

FIRST  CASE  :  THE  CURVE  C  RISES  CONSTANTLY  FROM  THE  POINT 
Ml  TO  THE  POINT  M2.  In  this  case  the  curve  c  rises  also  constantly 
from  Ml  to  M2,  and  except  at  the  points  M^  and  M2  it  is  always  above 
the  curve  C. 

This  case  is  represented  in  Fig.  60. 

SECOND  CASE:    BETWEEN  THE  POINTS  M1  AND  M2  THE  CURVE 

C  HAS  A  POINT  /  (FlG.  61),  OF  ABSCISSA  £  AND  ORDINATE  6,  HIGHER 
THAN  ALL  THE  OTHERS. 


1     X 


FIG.  62. 


According  to  the  first  theorem  of  Gibbs  and  of  Konovalow, 
this  point  /  is  an  indifferent  point;  under  the  pressure  TT,  at  the 
temperature  6,  the  liquid  mixture  and  the  saturated  mixed  vapor 
have  the  same  composition,  X=x=£.  The  curve  c  also  passes 
through  the  point  I  which  is,  on  this  curve,  a  point  above  all  the 
others;  outside  of  the  points  Mi}  I,  M2  the  curve  c  is  always  above  the 
curve  C. 


230  THERMODYNAMICS  AND   CHEMISTRY. 

THIRD  CASE:   BETWEEN  THE  POINTS  M±  and  M2  THE  CURVE  C 

(FlG.   62)    HAS   A   POINT  I,  OF  ABSCISSA   £  AND   ORDINATE   6,   LOWER 

THAN  ALL  THE  OTHERS.  In  this  case  the  curve  c  also  passes 
through  the  point  I,  which  is,  for  this  curve,  a  point  lower  than  all 
the  others;  outside  of  the  points  M1}  I,  M2  the  curve  c  is  always  above 
the  curve  C. 

Experiment  gives  us  numerous  examples  of  each  of  these  three 
cases. 

The  first  case  is  by  far  the  most  frequent;  it  occurs  with  the 
following  mixtures: 

Water-methyl  alcohol; 

Water-ethyl  alcohol; 

Water-acetic  acid; 

Water-butric  acid. 
Here  are  some  examples  of  the  second  case: 

Water-propyl  alcohol; 

Water-butryl  alcohol; 

Carbon  sulphide-ethyl  alcohol; 

Carbon  sulphide-ethyl  acetate; 

Carbon  tetrachloride-methyl  alcohol. 

The  first  two  mixtures  have  been  studied  by  Konovalow,1 
the  next  two  by  Brown,2  and  the  last  by  Thorpe.3 

According  to  Konovalow,  the  mixture  water-formic  acid  fur- 
nishes us  an  example  of  the  third  kind. 

196.  Distillation  of  a  mixture  of -two  volatile  liquids  under  con- 
stant pressure. — These  various  principles  are  going  to  permit  us 
to  study  the  phenomena  which  accompany  the  distillation  of  a 
mixture  of  two  liquids  under  constant  pressure.  In  the  still  the 
liquid  mixture  has  a  mixed  vapor  above  it ;  one  may  regard  this 
vapor  as  having  sensibly  the  composition  of  the  saturated  vapor 
in  equilibrium  with  the  mixed  liquid  for  the  conditions  of  tempera- 
ture and  pressure  which  reign  in  the  still.  At  every  instant  a 
portion  of  this  vapor  condenses  outside  of  the  still  and  a  new  mass 
of  liquid  vaporizes. 

1  D.  KONOVALOW,  Wiedemann's  Annal-en,  v.  14,  PD.  34  and  219,  1881. 

2  BROWN,  Quarterly  Journ.  of  the  Chem.  Soc.  of  London,  v.  80,  p.  529,  1881. 

8  THORPE,  Quarterly  Journ.  of  the  Chem.  Soc.  of  London,  v.  85,  p.  544,  1879. 


BIVARIANT  SYSTEMS.     THE  INDIFFERENT  POINT.    231 


The  following  proposition,  which  we  shall  take  as  starting-point, 
may  be  demonstrated: 

//  the  saturated  vapor  enclosed  by  the  still  has  not  the  same  com- 
position as  the  liquid  beneath  it,  the  boiling-point  of  the  liquid  rises 
as  the  distillation  proceeds. 

Take,  to  start  with,  a  liquid  mixture  which  comes  under  the  first 
of  our  three  cases. 

At  a  given  moment  the  liquid  contained  in  the  still  has  a 
certain  composition  X,  abscissa  of  a 
certain  point  M  (Fig.  63)  of  the  curve 
C;  the  temperature  within  the  still 
is  the  boiling-point  T  of  the  liquid  of 
composition  X,  that  is  to  say,  the  ordi- 
nate of  the  point  M;  on  the  curve  c 
there  is  a  point  m,  having  the  same 
ordinate  as  the  point  M;  the  abscissa 
x  of  this  point  m  gives  us  the  compo- 
sition of  the  vapor  which  fills  the  still 
at  the  instant  considered. 

The     composition     of    the    vapor 
differs   from   the    composition   of  the 


X      X  x'       X'     1       X 

FIG.  63. 

liquid;  the  boiling-point  of  the  liquid  contained  in  the  still  rises 
therefore,  with  the  effect  of  distillation.  After  a  certain  time  this 
boiling-point  has  assumed  a  value  T'  higher  than  T]  if  we  draw 
a  line,  parallel  to  X,  all  of  whose  points  have  the  ordinate  T',  this 
line  cuts  the  lines  C,  c  at  the  points  M',  m',  which  have  as  abscissae 
X'  and  x'  respectively;  X'  is  the  composition  of  the  liquid  in  the 
still,  x'  the  composition  of  the  enclosed  vapor,  at  the  moment 
when  the  boiling-point  becomes  T' . 

We  may  therefore  state  the  following  proposition: 
When  we  distil  under  constant  pressure  a  liquid  mixture  which 
is  among  the  first  of  our  three  cases,  the  composition  of  the  liquid 
remaining  in  the  still  and  the  composition  of  the  vapor  which  distils 
vary  always  in  the  same  way  and  tend  to  contain  only  the  less  vola- 
tile of  the  two  mixed  substances. 

Every  one  knows  that  the  above  is  true  for  the  distillation 
of  a  mixture  of  water  and  alcohol. 


232  THERMODYNAMICS  AND  CHEMISTRY. 

Quite  otherwise  is  the  action  for  a  mixture  which  is  of  the 
second  kind. 

197.  Mixtures  which  pass  over  entirely  by  distillation,  with- 
out variation  of  the  boiling-point. — Denote  always  by  £  and  6  the 
coordinates  of  the  indifferent  point  /. 

Reasoning  as  in  the  preceding  case,  we  may  establish  without 
difficulty  the  following  propositions:  When  a  liquid  mixture  is 
distilled  whose  initial  composition  corresponds  to  a  value  of  X  less 
or  greater  than  £,  the  boiling-point  constantly  rises  and  approaches 
0;  the  composition  of  the  liquid  in  the  still  and  the  composition 
of  the  vapor  in  it  vary  always  in  the  same  way,  so  as  to  approach  the 
common  composition  £. 

What  will  happen  at  the  moment  when,  the  liquid  and  vapor 
having  assumed  the  common  composition  £,  the  boiling-point 
will  have  reached  the  value  6?  Our  principle,  according  to  which 
the  boiling-point  should  constantly  rise  during  the  distillation,  is 
no  longer  applicable;  on  the  contrary,  as  the  vapor  within  the 
still  passes  over,  a  vapor  of  the  same  composition  may  replace  it 
without  either  the  composition  of  the  liquid  or  the  value  of  the 
boiling-point  changing.  When  the  composition  of  a  liquid  has 
reached  the  value  £  and  the  boiling-point  the  value  6,  there  is  estab- 
lished a  permanent  regime  of  distillation  in  which  the  boiling-point 
keeps  the  value  0,  while  the  vapor  which  distils  and  the  liquid  con- 
tained in  the  still  keep  a  constant  composition  £. 

This  regime  of  distillation  is  stable.  Thus,  if  any  cause  what- 
ever disturbs  it  in  one  direction  or  the  other,  the  action  of  the 
distillation  itself  will  tend,  as  we  have  seen,  to  reestablish  it.  A 
mixture  which,  as  the  mixture  of  formic  acid  and  water,  comes 
under  our  third  case,  may  have  a  permanent  regime  of  distillation* 
if  the  liquid  mixture  has  the  composition  £  which  accords  with  the 
indifferent  point,  the  vapor  has  the  same  composition ;  the  distilla- 
tion may  then  be  produced  without  change  of  composition  of  the 
liquid  or  of  the  vapor,  consequently  without  variation  of  the 
boiling-point,  which  remains  equal  to  6 ;  but  this  permanent  regime 
is  unstable;  if  any  circumstance  whatever  disturbs  it,  however 
slightly,  the  distillation  will  deviate  more  and  more  from  this 
regime.  In  fact,  reasoning  as  we  have  done  for  the  first  case,  we 
may  establish  the  following  proposition: 


BIVARIANT  SYSTEMS.    THE  INDIFFERENT  POINT.    233 

//  the  value  of  X  which  denotes  the  initial  composition  of  the  liquid 
is  less  than  £,  the  distillation  has  for  effect  to  increase  constantly  the 
proportion  of  the  fluid  1  in  the  vapor  and  in  the  liquid,  both  of  which 
tend  to  contain  this  substance  alone;  if,  on  the  contrary,  the  value  of 
X  which  indicates  the  initial  composition  of  the  liquid  is  greater  than 
£,  the  distillation  has  for  effect  to  increase  constantly  the  proportion 
of  the  fluid  2  in  the  vapor  and  in  the  liquid,  both  of  which  tend  to 
contain  this  substance  only. 

198.  These  mixtures  are  not  definite  compounds.  Researches 
of  Roscoe  and  Dittmar.  —  Let  us  return  to  the  permanent  and 
stable  regime  which  characterizes  our  second  case. 

Acted  upon  by  a  constant  pressure  n,  the  liquid  of  composition 
£  distils  at  a  constant  temperature  6.  furnishing  a  vapor  which 
has  an  identical  composition;  it  behaves,  therefore,  like  a  liquid 
body  of  definite  composition  which  vaporizes  and  whose  boiling- 
point  is  6  under  the  pressure  n.  Nevertheless,  if  one  were  tempted 
to  take  it  for  a  definite  compound,  there  is  a  property  which 
would  permit  us  to  distinguish  it  from  one.  The  composition 
of  a  definite  compound  does  not  change  with  the  pressure  to 
which  it  is  submitted;  on  the  contrary,  if,  instead  of  distilling 
one  liquid  mixture  under  a  pressure  n,  we  distil  it  under  a  differ- 
ent pressure,  n',  the  liquid  mixture  capable  of  passing  entirely 
over  in  the  distillation  without  change  of  composition  and  without 
variation  of  boiling-point  will  correspond  to  a  value  £'  of  X  which 
will  not,  in  general,  be  equal  to  £. 

A  solution  of  hydrochloric  acid,  subjected  to  atmospheric 
pressure,  begins  to  boil  at  a  temperature  which  rises  gradually  by 
distillation  to  110°  C.;  it  then  distils  in  constant  proportion  of 
water  and  hydrochloric  acid;  this  mixture  had  been  regarded 
by  Bineau  as  a  definite  chemical  compound  represented  by  the 
formula  HC1  •  8H2O.  Roscoe  and  Dittmar  1  did  not  consider 
the  matter  in  this  way,  and  they  showed  its  inaccuracy  by  causing 
hydrochloric  acid  solution  to  boil  under  various  pressures.  The 
boiling  attains  in  every  case  a  permanent  regime;  but  instead 
of  reproducing  constantly  the  supposed  hydrate  HC1-8H2O; 


and  DI-TTMAR,  Liebig's  Annalen,  v.  113,  p.  327,  1859;  Annales 
de  Chimie  et  de  Physique,  3d  S.,  v.  58,  p.  492,  1860. 


234 


THERMODYNAMICS  AND  CHEMISTRY. 


the  mixture  which  distilled  during  the  permanent  regime  had  a 
composition  variable  with  the  actual  pressure,  and  the  richer  in 
acid  as  the  pressure  rose. 

One  may  see  this  from  the  following  table,  where  TT  denotes 
the  pressure  in  centimetres  of  mercury,  and  £  the  number  of  grammes 
of  hydrochloric  acid  contained  in  1  gramme  of  the  solution  which 
has  a  constant  boiling-point  under  the  pressure  n. 


It 

* 

n 

f 

n 

I 

5 

0.232 

80 

0.202 

170 

0.188 

10 

0.229 

90 

0.199 

180 

0.187 

20 

0.223 

100 

0.197 

190 

0.186 

30 

0.218 

110 

0.195 

200 

0.185 

40 

0.214 

120 

0.194 

210 

0.184 

50 

0.211 

130 

0.193 

220 

0.183 

60 

0.207 

140 

0.191 

230 

0.182 

70 

0.204 

150 

0.190 

240 

0.181 

76 

0.2024 

160 

0.189 

250 

0.180 

When  one  distils  any  aqueous  solution  whatever  of  nitric  acid 
under  atmospheric  •  pressure,  there  always  comes  a  moment  when 
the  temperature  becomes  stationary  at  123°  C.  at  which  the  mix- 
ture passes  over  unchanged  by  the  distillation;  1  gramme  of  this 
mixture  contains  0.68  grammes  of  acid,  NO3H ;  if  a  mixture  richer 
in  acid  is  distilled,  very  concentrated  acid  first  passes  over,  a  por- 
tion of  this  acid  even  decomposes,  and  when  the  temperature 
reaches  123°  the  liquid  which  passes  over  and  that  remaining  have 
the  same  concentration;  when  a  weaker  acid  is  distilled,  water 
with  more  or  less  acid  passes  over  until  the  temperature  reaches 
123°. 

This  mixture,  which  has  a  fixed  boiling-point  and  passes  over 
as  a  whole  by  distillation,  is  not  a  definite  hydrate;  H.  Roscoe1 
has  shown  that  its  composition  varies  with  the  pressure  under 
which  the  distillation  takes  place;  1  gramme  of  this  mixture  con- 
tains 0.68  gr.  of  the  acid  NO3H  if  the  distillation  takes  place  under 
atmospheric  pressure;  if  the  distillation  takes  place  under  the 
pressure  of  7  centimetres  of  mercury,  this  gramme  of  mixture 


1  ROSCOE,  Liebig's  Annalen,  v.  116,  p.  203,  1860. 


BIVARIANT  SYSTEMS.    THE  INDIFFERENT  POINT.    235 


P: 


contains  but  0.667  gr.  of  acid;   it  contains  0.686  gr.  if  the  dis- 
tillation is  under  a  pressure  of  122  cm.  of  mercury. 

199.  Application  of  the  second  theorem  of  Gibbs  and  of  Kon- 
ovalow  to  mixtures  of  volatile  liquids. — The  study  of  the  tensions 
of  saturated  vapors  of  a  mixture  whose  composition  X  is  varied  at 
a  constant  temperature  T  offers  remarkable  similarities  in  all  points 
to  those  we  have  made  on  the  subject  of  boiling-points  under  a 
given  pressure. 

At  the  temperature  T,  let  Pl  and  P2  be  the  pressure  of  saturated 
vapors  of  the  liquids  1  and  2  taken  in  the  state  of  purity;  suppose 
also  that  the  liquid  1  be  more  volatile  than  the  liquid  2,  so  that 
Pt  exceeds  P2. 

The  mixture  of  composition  X  has,  at  the  temperature  con- 
sidered, a  tension  of  saturated  vapor  II. 
Take  a  point  N  (Fig.  64)  of  abscissa  X 
and  ordinate  II;  when  X  varies  from  0 
to  1  the  point  N  will  describe  a  curve  D 
joining  the  point  Nly  of  coordinates  0, 
Pv  to  the  point  Nt,  of  coordinates  1,  P2.  ] 

The  liquid  mixture  whose  composi- 
tion is  X  and  whose  vapor  tension  II 
is  in  the  presence  of  a  saturated  vapor 
of  composition  x;  the  point  n,  of  ab- 
scissa x  and  ordinate  77,  associated  with 
the  point  N  of  the  same  ordinate,  con- 
cludes the  representation  of  an  equilib- 
rium state  of  the  system.  When  X  varies  from  0  to  1,  z  varies  like- 
wise from  0  to  1  and  the  point  n  describes  a  curve  d  joining  N^  to  N2. 

There  are  three  principal  cases  to  distinguish: 

FIRST  CASE:  THE  CURVE  D  DESCENDS  CONSTANTLY  FROM  THE 
POINT  N1  TO  THE  POINT  N2. — In  this  case  the  curve  d  descends  like- 
wise constantly  from  the  point  Nl  to  the  point  N2;  the  curve  d  is, 
throughout  its  length,  below  the  curve  D. 

Fig.  64  shows  this  case. 

SECOND  CASE:  BETWEEN  THE  POINTS  JVj  AND  N2  (FiG.  65)  THE 

CURVE  D  HAS  A  POINT  7,  OF  ORDNATE  P  SMALLER  THAN  ALL  THE 

OTHERS. — According  to  the  second  theorem  of  Gibbs  and  of  Konovalow, 
this  point  is  an  indifferent  point  when  the  liquid  and  the  saturated 


p., 


o 


1     X 


FIG. 


236 


THERMODYNAMICS  AND  CHEMISTRY. 


vapor  have  the  same  composition  X=x=£,  so  that  the  point  I  is  also 
on  the  curve  d;  for  this  curve  it  is  an  ordinary  point  of  ordinate  smaller 
than  all  the  others;  outside  of  the  points  Nlt  /,  N2  the  curve  d  is  en- 
tirely below  the  curve  D. 

n 


N3 


I  l 

FIG.  65. 


1    X 


FIG.  66. 


THIRD  CASE:  BETWEEN  THE  POINTS  #t  AND  N2  (Fio.  66)  THE 

CURVE  D  HAS  A  POINT  I,  OF  ORDINATE  %  GREATER  THAN  ALL  THE 

OTHERS. — The  curve  d  also  passes  through  this  point  I,  which  is  an 
indifferent  point  (X=x  =  £),  and  has  there  an  ordinate  greater  than 
all  the  others.  Outside  of  the  points  N1 1,  N2  the  curve  d  is  constantly 
below  the  curve  D. 

200.  Distillation  of  a  mixture  of  two  liquids  at  constant  tem- 
perature.— This  conclusion,  furnished  by  thermodynamics,  will 
permit  us  to  discuss  the  phenomena  of  distillation  which  are  pro- 
duced when,  without  changing  the  temperature,  the  vapor  emitted 
by  the  mixed  liquid  is  constantly  aspirated,  provided  that  we  make 
use  of  the  following  proposition,  furnished  likewise  by  thermo- 
dynamics : 

When  the  liquid  and  the  vapor  which  it  emits  have  not  the  same 
composition,  distillation  may  not  be  produced  at  a  constant  tempera- 
ture, unless  the  pressure  of  the  vapor  constantly  diminishes. 

By  the  same  sort  of  reasoning  we  used  concerning  distillation 
under  constant  pressure,  we  may  without  difficulty  establish  the 
following  results: 

FIRST  CASE. — During  distillation  the  vapor  tension  constantly 


BIVARIANT  SYSTEMS.     THE  INDIFFERENT  POINT.    237 

decreases  and  approaches  the  tension  of  saturated  vapor  P2  of  the 
substance  2  taken  in  a  state  of  purity;  the  proportion  of  the  sub- 
stance 1  constantly  diminishes,  both  in  the  liquid  and  in  the  vapor 
form;  both  tend  to  be  no  longer  formed  otherwise  than  of  substance  2. 

SECOND  CASE. — Whether  the  value  of  X  which  represents  the 
initial  composition  of  the  liquid  be  less  than  £  or  higher  than  £,  the 
vapor  tension  constantly  decreases  and  approaches  P;  the  composi- 
tion X  of  the  liquid  and  the  composition  x  of  the  vapor  both  vary  in 
the  same  way  and  approach  £.  When  the  vapor  tension  reaches  the 
value  P  there  is  established  a  permanent  regime  of  distillation;  the 
vapor  tension  changes  no  more;  the  vapor  distilled  and  the  liquid  not 
distilled  conserve  a  constant  composition  £.  This  permanent  regime 
is  stable. 

THIRD  CASE. — The  system  may  possess  a  permanent  regime  of 
distillation,  under  the  invariable  pressure  2,  the  liquid  and  the  vapor 
having  the  same  constant  composition  £;  but  this  regime  is  unstable. 

If  the  value  of  X  which  represents  the  initial  composition  of  the 
liquid  is  less  than  £,  while  the  pressure  diminishes  and  approaches 
the  tension  of  saturated  vapor  P±  of  the  fluid  I,  the  composition  of 
the  liquid  and  the  composition  of  the  vapor  constantly  vary  in  the  same 
direction;  these  two  fluids  tend  to  be  formed  of  the  substance  1  alone. 

If  the  value  of  X  which  represents  the  initial  composition  of  the 
liquid  is  higher  than  £,  while  the  pressure  diminishes  and  approaches 
the  tension  of  saturated  vapor  P2  of  the  substance  2,  the  composition 
of  the  liquid  and  the  composition  of  the  vapor  constantly  change  in  the 
same  direction;  these  two  fluids  tend  to  be  formed  of  the  same  substance 
2  alone. 

In  our  second  case,  the  mixture  of  concentration  £  which  has, 
at  the  temperature  considered,  a  definite  vapor  tension  and  passes 
over  unaltered  by  distillation,  may  be  confused  with  a  definite 
compound;  this  confusion  may  be  readily  obviated  if  it  is  noticed 
that  the  composition  of  the  mixture  having  these  properties  de- 
pends on  the  temperature. 

The  evaporation,  at  ordinary  temperature,  of  an  aqueous  solu- 
tion of  hydrochloric  acid  always  furnishes,  after  a  certain  time, 
a  mixture  of  constant  vapor  tension  and  of  constant  composition, 
which  Bineau  had  considered  as  a  definite  hydrate,  represented 


238  THERMODYNAMICS  AND  CHEMISTRY. 

by  the  formula  HC1  •  6H20 ;  Roscoe  and  Dittmar  1  have  shown 
that  the  composition  of  this  mixture  varies  with  the  temperature 
at  which  evaporation  takes  place. 

20 1.  Relation  between  distillation  at  constant  pressure  and 
distillation  at  constant  temperature. — Between  the  permanent 
regime  which  may  be  established  when  a  mixture  is  distilled  at 
constant  pressure  and  the  permanent  regime  which  may  be  estab- 
lished when  evaporation  takes  place  at  constant  temperature, 
there  exists  a  relation. 

Suppose  that,  under  the  pressure  P,  we  may  observe  a  condi- 
tion of  indifferent  equilibrium  where  the  liquid  mixture  and  the 
saturated  vapor  have  the  same  composition  £;  the  boiling-point  0 
of 'the  mixture  of  composition  £  is  maximum  or  minimum  among 
the  boiling-points  which  the  liquid  mixture  may  have  under  con- 
stant pressure  P. 

At  the  constant  temperature  6  the  liquid  mixture  of  composi- 
tion £  will  give  off  a  saturated  vapor  of  same  composition  with 
which  it  will  be  in  indifferent  equilibrium ;  the  tension  of  the  satu- 
rated vapor  of  this  mixture  of  composition  £  will  have  a  value  P; 
this  value  must  be  a  maximum  or  minimum  among  the  tensions 
of  saturated  vapors  which  the  liquid  mixture  may  have  at  the 
constant  temperature  6. 

The  following  propositions  may  be  demonstrated: 

If  the  temperature  6  is  a  maximum  among  the  boiling-points 
which  the  liquid  mixture  may  have  when  its  composition  is  varied 
by  keeping  constant  the  pressure  P,  the  pressure  P  will  be  a  mini- 
mum among  the  tensions  of  saturated  vapor  from  the  liquid  mix- 
ture when  its  composition  is  varied,  leaving  constant  the  tempera- 
ture 6. 

If  the  temperature  0  is  a  minimum  among  the  boiling-points 
possessed  by  the  liquid  mixture  when  its  composition  is  varied 
keeping  the  pressure  P  constant,  the  pressure  P  will  be  a  maxi- 
mum among  the  tensions  of  saturated  vapor  from  the  liquid  mix- 
ture when  its  composition  is  varied,  leaving  the  temperature  0 
constant. 

1  ROSCOE  and  DITTMAR,  Liebig's  Annalen,  v.  112,  p.  327,  1859;  Annales 
de  Chimie  et  de  Physique,  3d  S.,  v.  58,  p.  492,  1860. 


B1VARIANT  SYSTEMS.    THE  INDIFFERENT  POINT.   239 

The  first  of  these  two  propositions  is  evidently  the  equivalent 
of  the  following: 

Suppose  that  a  liquid  mixture  is  being  distilled  under  constant 
pressure  P;  there  comes  a  moment  when  the  distillation  lets  pass  over 
a  vapor  of  constant  composition  £,  the  boiling-point  becoming  fixed 
at  the  value  0,  constant  from  this  mo/nent;  conversely,  if  this  mixture 
is  evaporated  at  the  temperature  6,  the  tension  of  the  saturated  vapor 
wiU  finally  become  fixed  at  the  value  P,  and  the  evaporation  will 
furnish  a  vapor  of  constant  composition,  still  equal  to  £. 

This  law  has  been  experimentally  established  by  Roscoe  and 
Dittmar  *  by  studying  mixtures  of  water  and  hydrochloric  acid. 

1  ROSCOE  and  DITTMAK,  loc.  tit. 


CHAPTER  XII. 


BIVARIANT  SYSTEMS   (Continued}.     TRANSITION  AND    EUTEXIA. 

202.  Common  point  to  the  solubility  curves  of  two  hydrates. 
Three*  cases  to  distinguish. — Suppose  that  a  solution  of  a  salt  in 
water  may  give  two  different  solid  precipitates,  both  of  definite 
composition:  for  example,  a  salt,  anhydrous  or  hydrated,  and  ice, 
or  an  anhydrous  salt  and  a  hydrated  salt,  or  again  two  different 
hydrated  salts ;  let  us  operate  under  a  pressure  given  once  for  all, 
and  let  us  ask  ourselves  if,  under  this  pressure,  we  may  observe 
a  system  in  equilibrium  containing  at  once  the  solution  and  the 
two  precipitates. 

When  the  two  precipitates  coexist  in  contact  with  the  solution, 
the  system,  always  formed  of  two  independent  components,  is 
divided  into  three  phases;  it  is  no  longer  bivariant,  but  mono  vari- 
ant; in  general  it  cannot  be  in  equi- 
librium under  the  pressure  considered, 
except  at  a  particular  temperature 
which  we  shall  denote  by  0. 

It  is  not  difficult  to  define  in  an 
exact  manner  the  temperature  6. 

Let  a  and  b  be  our  two  precipi- 
tates. Under  the  pressure  considered, 
the  precipitate  a  has  a  solubility 
curve,  the  curve  Ca  (Fig.  67) ;  in  order 
that  the  solution  be  in  equilibrium  in 
contact  with  the  substance  a,  it  is  ne- 
cessary and  sufficient  that  the  repre- 
sentative point,  which  has  the  temperature  for  abscissa  and  the  con- 

240 


0 
FIG.  67. 


BIVARIANT  SYSTEMS.    TRANSITION  AND  EUTEXIA.  241 

centration  of  the  solution  for  ordinate,  be  on  the  line  Ca.  The 
precipitate  b  likewise  has  a  solubility  curve  C&;  in  order  that  the 
solution  remain  in  equilibrium  in  contact  with  the  precipitate  b, 
it  is  necessary  and  sufficient  that  the  representative  point  be  on 
the  curve  C&. 

It  is  therefore  clear  that  for  a  solution  to  remain  in  equilibrium, 
under  the  pressure  considered,  in  contact  with  the  two  precipitates  a 
and  b,  it  is  necessary  and  sufficient  that  the  temperature  has  the  value 
0,  and  the  concentration  of  the  solution  the  value  2,  0  and  I  being  the 
coordinates  of  the  point  a>  common  to  the  two  solubility  curves  Ca 
and  Cb> 

When  the  representative  point  is  elsewhere  than  at  CD,  it  is 
impossible  for  our  bivariant  system  to  remain  in  equilibrium;  it 
must  be  transformed  until  the  complete  disappearance  of  one  of 
its  phases  takes  place.  What  laws  govern  these  transformations? 
To  determine  these  laws,  we  must  distinguish  three  cases,  as  follows: 

FIRST  CASE. — The  solution  of  concentration  I  contains  more 
water  than  either  of  the  two  precipitates  a  and  b. 

This  case  may  also  be  thus  defined: 

The  two  curves  Ca  and  C&  which  intersect  at  the  point  to  are  the 
lower  branches  of  the  solubility  curves  of  the  two  precipitates  a  and  b. 

SECOND  CASE. — The  solution  of  concentration  2  contains  less 
water  than  either  of  the  two  precipitates  a  and  b. 

We  may  state  this  case  in  the  following  way: 

The  two  curves  Ca,  C&,  intersecting  at  the  point  CD,  are  the  upper 
branches  of  the  solubility  curves  of  the  two  precipitates  a  and  b. 

THIRD  CASE. — The  solution  of  concentration  I  contains  less  water 
than  the  precipitate  a  and  more  than  the  precipitate  b. 

Otherwise  stated  this  case  becomes: 

The  part  of  the  curve  Ca  which  passes  through  the  point  CD  is  the 
upper  branch  of  the  solubility  curve  of  the  precipitate  a;  the  part  of 
the  curve  C&  passing  through  the  point  CD  is  the  lower  branch  of  the 
solubility  curve  of  the  precipitate  b. 

This  third  case  offers  peculiarities  which  sharply  distinguish 
it  from  the  first  two;  on  the  contrary,  the  properties  of  the  first 
two  are  so  analogous  that  it  will  be  sufficient  for  us  to  study  one 
of  them,  the  first  for  example. 

203.  Transition-point— Of  the  two  curves  Ca,  C6,  which  inter- 


242  THERMODYNAMICS  AND  CHEMISTRY. 

sect  at  the  point  a>,  there  is  one  of  them,  followed  from  left  to  right, 

which  rises  more  sharply  or  descends 
less  sharply  than  the  other;  suppose 
this  to  be  the  curve  Ca.  Then  the 
branch  Ca  of  this  curve  (Fig.  68) 
which  corresponds  to  temperatures 
less  than  6  is  below  the  corresponding 
branch  Cb  of  the  curve  Cb;  on  the 
contrary,  the  branch  Car  of  the 
curve  Ca  which  relates  to  tempera- 
tures above  6  is  above  the  corre- 


sponding branch  Cb  of  the  curve  C6. 
FlG-  68-  At  a  temperature  less  than  6  can 

the  solution  remain  in  equilibrium  in  contact  with  the  precipitate 
6?  In  order  that  the  solution  remain  in  equilibrium  in  contact 
with  the  precipitate  b,  it  is  necessary  in  the  first  place  that  it  may 
neither  dissolve  nor  abandon  a  certain  mass  of  this  precipitate, 
which  requires  that  the  representative  point  lie  on  the  line  C&; 
but  this  is  not  sufficient;  it  is  further  necessary  that  the  solution 
may  not  give  birth  to  a  certain  mass  of  the  precipitate  a,  which 
requires  that  the  representative  point  lie  below  the  line  Ca,  since 
this  line  is  the  solubility  curve  of  a  substance  less  rich  in  water 
than  the  solution.  It  is  therefore  evident  that  at  temperatures 
below  6  the  precipitate  b  cannot  remain  in  equilibrium  in  contact 
with  the  solution. 

It  may  be  shown  similarly  that  at  temperatures  above  6  the 
precipitate  a  cannot  remain  in  equilibrium  in  contact  with  the  solu- 
tion. 

As  to  the  mixture  of  the  two  solid  precipitates,  exempt  from  liquid 
solution,  it  remains  necessarily  in  equilibrium  at  temperatures  near 
to  6,  whether  they  are  above  or  below  6.  If,  therefore,  this  mixture 
of  two  solid  substances,  each  of  which  is  less  rich  in  water  than  the 
solution. of  concentration  2,  underwent  aqueous  fusion,  the  solu- 
tion produced  would  have  a  concentration  greater  than  2;  the 
temperature  being  close  to  0,  the  representative  point  of  this  solu- 
tion would  lie  above  the  two  curves  Ca  and  C&,  so  that  the  solution, 
supersaturated  by  each  of  the  substances  a  and  b,  could  not  re- 
main in  equilibrium. 


BIVARIANT  SYSTEMS.    TRANSITION  AND  EUTEXIA-  243 


204.  Various  examples :  sodium  sulphate. — There  are  numerous 
opportunities  in  practice  to  apply  these  principles. 

The  case  known  for  the  longest  time  is  that  furnished  by  so- 
dium sulphate,  carefully  studied  by  Loewel.1  Hydrated  sodium 
sulphate,  Na.2SO4  •  10H2O,  dissolves  with  absorption  of  heat,  so 
that  the  branch  of  the  solubility  curve  along  which  the  solution  is 
less  concentrated  than  the  hydrate  (the  only  one  known)  rises 
from  left  to  right  following  Caaj  (Fig.  69).  Anhydrous  sodium  sul- 
phate, Na^SC^,  dissolves  with  liberation 
of  heat,  so  that  the  solubility  curve 
for  this  substance  descends  from  left 
to  right  following  wCb.  These  two 
curves  intersect  in  a  point  to,  whose  z 
abscissa  corresponds  sensibly  to  the 
temperature  +33°  C.  From  what 
precedes,  at  temperatures  less  than 
+  33°  C.  the  only  true  equilibrium 
which  can  be  observed  is  the  equilib- 
rium between  the  solution  and  the 
anhydrous  sodium  sulphate;  this  is,  in 
fact,  what  experiment  shows.  By  means  of  a  phenomenon  of 
false  equilibrium  one  may  observe  systems  in  which  a  solution, 
supersaturated  with  respect  to  hydrated  sodium  sulphate,  is  in 
equilibrium  in  the  presence  of  anhydrous  sodium  sulphate;  the 
states  of  equilibrium  thus  obtained  are  represented  by  various 
points  of  the  line  Cb'w. 

205.  Thorium  sulphate. — Roozboom  has  called  attention  to 
a  case  2  analogous   to   that   of   sodium   sulphate,  but  where   the 
phenomena  of  supersaturation  are  produced  with  an  exceptional 
facility;  this  case  is  that  of  thorium  sulphate. 

Thorium  sulphate  with  9  molecules  of  water,  Th(S04)2-9H,0, 
dissolves  with  absorption  of  heat  and  corresponds  to  a  solubility 
curve  CaajCa'  (Fig.  70)  rising  from  left  to  right;  on  the  con- 
trary, thorium  sulphate  with  4  molecules  of  water,  Th(SO4)2-4H2O, 

1  LCEWEL,  Annales  de  Chimie  et  de  Physique,  3d  S.,  v.  29,  p.  62,  1850. 

2  ROOZBOOM,  Archives  neerlandaises  des  Sciences  exactes  et  naturettes,  v.  24; 
Zeitschrift  fur  physikalische  Chemie,  v.  5,  p.  198,  1890. 


e          T 
FIG.  69. 


244 


THERMODYNAMICS   AND   CHEMISTRY. 


dissolves  with  liberation  of  heat  and  corresponds  to  a  solubility 

curve  Cb'cuCb  which  descends  from 
left  to  right;  these  two  curves  in- 
tersect at  a  transition-point  10  whose 
abscissa  corresponds  to  the  tem- 
perature +  43°  C. 

The  two  branches  Cato,  C^ 
alone  correspond  to  states  of  true 
equilibrium;  nevertheless  the  line 
Caa>  might  be  prolonged  beyond 
the  point  oj,  to  the  point  Ca',  whose 
abscissa  Ta  corresponds  to  the 
temperature  +55°  C.,  but  the 
segment  a>Ca'  represents  solutions 
supersaturated  with  respect  to  the 
of  water;  and  the  line  GIOJ  might 

whose 


o      T6         e      Ta 
FIG.  70. 

hydrate  with  4   molecules 

have  been  extended,  beyond  the  point  aj,  to  the  point 

abscissa   T&  corresponds  to  the  temperature  +17°  C.,  while  the 

segment  C&'o>  represents  solutions  supersaturated  with  respect  to 

the  hydrate  of  9  molecules  of  water. 

When  we  meet  thus  a  point  common  to  two  solubility  curves 
of  two  hydrates  of  the  same  salt,  and  if  at  this  point  the  solution 
is  richer  in  water  than  either  of  the  two  hydrates  or  less  rich  in 
water  than  either  of  the  two  hy-  s 
drates,  we  shall  say  that  this  point 
is  a  transition-point. 

206.  Eutectic  point. — Things 
take  place  quite  differently  for  the 
last  of  the  three  cases  which  we 
have  enumerated ;  in  this  case  the 
saturation  curve  CaCa'  of  the  pre- 
cipitate a  and  the  saturation  curve 
CbCb  of  the  precipitate  b  intersect 
at  the  point  w(Fig.  71)  of  abscissa 
0  and  ordinate  I',  the  solution  of 
concentration  2  is  less  rich  in 
water  than  the  precipitate  a  and  richer  in  water  than  the  pre- 
cipitate b.  Otherwise  expressed,  the  branch  of  the  curve  CaCa' 


e 
FIG.  71. 


BIVARIANT  SYSTEMS.    TRANSITION  AND  EUTEXIA.  245 

which  passes  through  the  point  <u  belongs  to  the  upper  branch  of 
the  solubility  curve  of  the  hydrate  a;  on  the  contrary,  the  branch 
of  the  curve  C&Cb'  which  passes  through  the  point  w  belongs  to 
the  lower  branch  of  the  solubility  curve  of  the  hydrate  6. 

To  fix  our  ideas,  we  shall  suppose  that  the  two  precipitates  dis- 
solve with  absorption  of  heat;  it  follows  from  what  we  have  seen 
in  the  preceding  chapter  (Art.  188)  that  the  curve  Ca'Ca  descends 
from  left  to  right,  while  the  curve  C&'Cf,  rises  from  left  to  right. 

Outside  of  the  point  cu,  we  cannot  observe  in  equilibrium  the 
system  divided  into  three  phases;  but  it  may  be  that  we  may 
observe,  in  equilibrium,  a  system  divided  into  two  phases,  which 
may  occur  in  three  wa>s: 

1°.  The  system  may  be  formed  by  the  substance  a  in  contact 
with  a  solution; 

2°.  The  system  may  consist  of  the  substance  6  with  a  solution; 

3°.  The  system  may  be  composed  of  the  substances  a  and  b  in 
the  absence  of  any  solution. 

In  order  that  a  system  formed  of  the  precipitate  a  in  contact 
with  a  solution  remain  in  equilibrium,  it  is  necessary,  in  the  first 
place,  that  the  solution  be  saturated  with  the  substance  a,  or,  in 
other  terms,  that  the  representative  point  lie  on  the  curve  CaCa'; 
but  this  is  not  sufficient;  it  is  further  necessary  that  the  solution 
may  not  give  birth  to  the  precipitate  b,  that  it  be  not  saturated 
with  respect  to  this  precipitate,  consequently  that  the  representa- 
tive point  does  not  lie  above  the  curve  C&C&' ;  whence  the  following 
conclusion : 

In  order  that  a  system  enclosing  the  precipitate  a  and  a  solution 
may  remain  in  equilibrium,  it  is  necessary  and  sufficient  that  the 
representative  point  lie  on  the  branch  wCa  of  the  curve  CaCa',  starting 
from  the  point  cj  and  extending  to  the  right  of  this  point. 

In  order  that  a  system  formed  of  the  precipitate  b  in  contact 
with  a  solution  be  in  equilibrium,  it  is  necessary  in  the  first  place 
that  the  solution  be  saturated  with  the  substance  6,  that  is  to 
say,  that  the  representative  point  lie  on  the  line  C&C&' ;  and  further- 
more it  is  also  requisite  that  the  precipitate  a  cannot  be  gen- 
erated in  the  solution,  and,  as  the  saturated  solutions  of  the  sub- 
stance a  are  represented  by  the  various  points  of  the  plane  located 
below  the  line  CaCa',  it  is  necessary  that  the  representative  point 


246  THERMODYNAMICS  AND  CHEMISTRY. 

considered  does  not  lie  below  this  line  CaC0';  whence  the  following 
conclusion : 

For  a  system  which  contains  the  precipitate  b  and  a  solution  to 
remain  in  equilibrium,  it  is  necessary  and  sufficient  that  the  repre- 
sentative point  be  on  the  branch  ajCb  of  the  line  C^Cb ',  starting  from 
the  point  a)  and  extending  to  the  right  of  this  point. 

Finally,  let  us  consider  a  system  which  contains  the  two  solid 
precipitates  a  and  6 ;  is  it  going  to  remain  in  equilibrium  or  undergo 
aqueous  fusion? 

Imagine  that  a  part  of  the  two  hydrates  undergoes  aqueous 
fusion  and  gives  rise  to  a  solution;  this  solution  cannot  be  richer  in 
water  than  the  hydrate  a,  and  consequently  than  the  two  hydrates; 
it  cannot,  either,  be  less  rich  in  water  than  the  hydrate  b,  and  there- 
fore than  the  two  hydrates ;  it  has  necessarily  a  composition  inter- 
mediate between  that  of  the  hydrate  a  and  that  of  the  hydrate  b. 

Suppose  the  temperature  less  than  6;  the  points  situated  below 
the  line  C\[u  represent  solutions  supersaturated  with  respect  to 
the  precipitate  a;  the  points  situated  between  the  lines  C&'w  and 
Ca'(D  represent  solutions  supersaturated  with  respect  to  the  two 
precipitates  a  and  6;  the  points  situated  above  the  curve  Cafa) 
represent  solutions  supersaturated  with  respect  to  the  precipitate 
b;  whatever  the  composition  of  the  solution  formed,  it  is  super- 
saturated with  respect  to  at  least  one  of  the  two  precipitates  a 
and  6,  so  that  it  will  give  back  a  solid  precipitate,  and  that  until 
the  solution  has  entirely  disappeared;  aqueous  fusion  is  therefore 
impossible  at  temperatures  less  than  6. 

At  temperatures  less  than  6  a  solid  mixture  of  the  two  precipitates 
a  and  b  cannot  undergo  aqueous  fusion;  a  solution  of  any  compo- 
sition solidifies  and  forms  a  mixture  of  the  two  precipitates. 

Consider  next  a  temperature  higher  than  6 ',  can  the  two  solid 
precipitates  a  and  b  remain  in  equilibrium  at  this  temperature 
without  undergoing  aqueous  fusion?  Such  an  equilibrium,  sup- 
posing it  possible  to  realize,  would  be  unstable.  Thus  imagine 
that  a  very  small  part  of  the  mixture  of  the  two  solids  undergoes 
aqueous  fusion  and  gives  a  drop  of  solution. 

If  the  point  representing  this  solution  is  above  a>Ca,  the  solu- 
tion may  dissolve  the  precipitate  a;  if  it  is  below  wCb,  the  solution 
may  dissolve  the  precipitate  6;  therefore  whatever  be  the  position 


BIVARIANT  SYSTEMS.    TRANSITION  AND  EUTEXIA.  247 


Of  the  representative  point,  there  is  at  least  one  of  the  two  pre- 
cipitates that  the  solution  may  dissolve.  Hence,  from  the  instant 
a  drop  of  solution  is  produced  in  the  system  at  a  temperature 
higher  than  0,  equilibrium  cannot  be  reestablished  in  the  system 
unless  at  least  one  of  the  two  precipitates  has  passed  entirely  into 
the  so-ution. 

At  temperatures  above  0  a  system  containing  the  two  solid  sub- 
stances a  and  b,  in  any  proportion,  passes  into  the  liquid  state  until 
at  least  one  of  the  two  solid  bodies  has  disappeared. 

The  temperature  0  is  the  fusing-point  of  a  system  which  contains 
the  two  solids  a  and  b  at  once. 

207.  Formation  of  the  eutectic  mixture. — We  shall  show  some 
new  properties  of  the  temperature  0  and  of  the  concentration  2 
by  examining  the  following  question: 

At  a  temperature  above  6  a  solution  is  taken  which  is  saturated 
neither  with  the  substance  a  nor  with  the  substance  b,  which  never- 
theless is  in  equilibrium;  the  temperature  is  gradually  lowered;  what 
are  the  precipitates  from  the  solution  f 

There  are  two  cases  to  distinguish  according  as  the  initial  con- 
centration of  the  solution  is  greater  or  less  than  2. 

FIRST  CASE. — The  initial  concentration  s  is  greater  than  2. 

The  representative  point  for  the 
initial  state  of  the  solution  is  a  point 
M  (Fig.  72)  higher  than  the  point  a). 

When  the  temperature  is  lowered, 
starting  from  its  initial  value  T, 
the  representative  point  remains  at 
first  in  the  region  included  between 
the  lines  wCa  and  ajCb ;  the  solution, 
not  being  saturated  either  with  the 
substance  a  or  with  the  substance 
6,  gives  no  precipitate  and  its  con- 
centration remains  invariable;  the 


T  T 


o         e  / 

PIG.  72. 

representative  point  describes  a  parallel  Mm  to  the  straight  line  OT. 
When  the  temperature  is  lowered  to  a  certain  value  t,  the 
representative  point  is  situated  on  the  curve  cuCb  at  m;  the  solu- 
tion is  then  saturated  with  the  substance  6. 

The  temperature  falling  below  t,  the  solution  deposits  a  certain 


248 


THERMODYNAMICS  AND  CHEMISTRY. 


amount  of  the  substance  6,  so  as  to  remain  saturated  with  this 
substance;  the  representative  point  describes  the  part  maj  of  the 
line  ajCb' 

At  the  instant  when  the  temperature,  always  falling,  attains 
the  value  6,  the  representative  point  is  at  to  and  the  concentration 
has  the  value  I. 

If  we  lower  the  temperature  by  a  small  fraction  below  d,  the 
solution  solidifies ;  the  solid  deposit  which  it  furnishes  is  not  homo- 
geneous; it  is  formed  by  a  juxtaposition  of  particles  of  solids  a 
and  6;  but  its  mean  composition  is  well  determined;  it  is  the 
same  as  the  composition  of  the  solution  of  concentration  I  which 
has  furnished  it. 

SECOND  CASE. — The  initial  composition  of  the  solution  is  less 
than  I. 

The  representative  point  for  the  initial  state  of  the  solution 

is  a  point  M  (Fig.  73)  lower  than 
the  point  a>. 

When  the  temperature,  start- 
ing from  the  initial  value  T,  com- 
mences to  fall,  the  representative 
point  remains,  at  first,  included 
between  the  lines  cuCa  and  a>Cb; 
the  solution,  being  saturated 
neither  with  the  substance  a  nor 
with  b,  gives  no  precipitate;  its 
concentration  remains  constant  and 
the  representative  point  describes 
a  line  Mm  parallel  to  OT. 
This  parallel  meets  in  a  point  m,  of  abscissa  t,  the  line  ajCa',  at 
the  moment  when  the  temperature  reaches  the  value  t  the  solu- 
tion becomes  saturated  with  the  substance  a.  If  the  tempera- 
ture is  brought  lower  than  t,  the  solution  gives  up  some  of  the  sub- 
stance a  in  the  solid  state  so  as  to  remain  saturated  with  this 
substance;  the  representative  point  describes  the  segment  mco 
of  the  line  a)Ca> 

The  instant  the  temperature  attains  the  value  6  the  repre- 
sentative point  is  at  aj  and  the  concentration  of  the  solution  has 
the  value  I. 


e          t 

FIG.  73. 


T   T 


BIVARIANT  SYSTEMS.     TRANSITION  AND  EUTEXIE.    249 


If  the  temperature  falls  below  6,  the  solution  solidifies;  it  gives 
a  mixed  precipitate,  formed  of  particles  of  the  substance  a  and  of 
b,  whose  average  composition  is  well  determined;  this  composi- 
tion is  that  of  the  solution  of  concentration  2. 

If  a  solution  of  any  initial  composition  whatever  is  cooled,  it  lets 
first  precipitate  either  the  substance  a  or  the  substance  b,  in  the  pure 
state;  but  at  the  instant  the  temperature,  in  falling,  passes  through  6, 
the  solution  solidifies;  the  solid  obtained  is  not  homogeneous;  it  is 
formed  by  the  juxtaposition  of  particles  of  the  substances  a  and  b; 
but  its  mean  composition  is  perfectly  determined,  being  identical 
with  that  of  a  solution  of  concentration  2. 

Guthrie  has  given  the  name  eutectic  mixture  to  the  solid  magma 
obtained  in  these  conditions;  the  point  w  is  a  eutectic  point. 

208.  Particular  case:  ice  and  anhydrous  salt. — The  phenom- 
ena whose  laws  we  have  just  sketched  were  first  studied  by 
taking  for  substance  a  ice  and  for  substance  6  a  salt,  anhydrous 
or  hydrated;  the  cooling  mixtures  that  are  obtained  by  mixing 
ice  with  a  salt,  such  as  sea-salt  and  saltpetre,  had  already  been 
noticed  by  physicists  near  the  end  of  the  eighteenth  century,  and 
the}-  had  learned  some  of  the  properties  of  the  point  to;  it  was 
known  that  the  salt  could  no  longer  make  the  ice  melt  when  the 
temperature  is  lower  than  6,  temperature  at  which  ice  is  formed  in 
the  midst  of  a  solution  saturated  with  the  salt,  or,  in  other  terms, 
abscissa  of  the  point  of  intersection  of  the  curve  wCa  of  the  freezing-- 
points and  of  the  solubility  curve  wC^  for  the  salt  considered.  The 
temperature  0  is  thus  the  lower  limit  of  the  temperatures  which 
may  be  produced  by  means  of  a  cooling  mixture  formed  of  ice 
mixed  with  the  salt  considered. 


Salt. 

e 

2 

Potassium  sulphate 

-  1°.9 

0.10 

Potassium  nitrate     

-  2  .8 

0.13 

Potassium  chloride   

-10  .9 

0.30 

Ammonium  nitrate  

-16  .7 

0.45 

Sodium  chloride  

-21  .3 

0.33 

De  Coppet,1  first,  and  then  Guthrie  and  other  observers  under- 
took to  determine  with  accuracy  the  temperature  6  for  a  number 
1  DE  COPPET,  Bulletin  de  la  Sotiett  Vaudoise,  2d  S.,  v.  n,  p.  1,  1871. 


250  THERMODYNAMICS  AND   CHEMISTRY. 

of  salts;  some  of  these   temperatures   are  given  in  the  table; 
there  is  joined  to  them  the  value  of  the  ordinate  2,  of  the  point  a). 

209.  Non-existence  of  the  cryohydrates. — Brought  to  a  tem- 
perature below  6,  the  solution  solidifies;  the  magma  obtained  is 
not  a  definite  compound;  particles  of  ice  are  mixed  with  salt 
crystals;  but  its  composition  is  quite  definite;  it  is  identical  with 
that  of  a  solution  of  concentration  2. 

Guthrie,1  who  had  observed  with  much  care  the  formation  of 
this  solid  of  invariable  composition,  had  at  first  regarded  it  as 
a  definite  hydrate  for  which  6  would  be  the  aqueous  fusing-point; 
to  this  hydrate  he  gave  the  name  cryohydrate. 

One  might  decide  between  this  opinion  and  the  preceding  theory 
by  repeating  Guthrie's  experiments  at  a  pressure  quite  different 
from  atmospheric  pressure;  the  composition  of  the  solid  fur- 
nished by  the  solution  at  the  instant  of  solidification,  in  Guthrie's 
opinion,  should  be  independent  of  the  pressure  exerted  upon  the 
system;  in  the  opinion  advanced  here,  on  the  contrary,  the  com- 
position would  in  general  depend  upon  the  pressure. 

In  default  of  these  experiments  which  would  be  conclusive, 
but  which  the  extreme  smallness  of  the  variation  to  be  recognized 
would  render  extremely  difficult,  other  arguments  may  be  invoked 
against  the  existence  of  the  cryohydrates. 

In  the  first  place,  no  simple  formula  can  be  found  for  these 
substances;  for  example,  here  are  some  of  the  formulae  proposed 
by  Guthrie: 

Na2SO4+166H20; 
KC1O3  +  222H2O; 
Ba(NOs)3  +  259HO; 
A1NH4(SO4)2  +  261H2O. 

In  the  second  place,  a  microscopic  examination  of  the  pretended 
cryohydrates,  either  in  natural  light 2  if  the  salt  is  colored,  or  in 
polarized  light  if  the  salt  is  colorless  but  non-isotropic,  shows 
that  these  substances  are  in  no  way  homogeneous  and  that  they 
are  formed  of  crystals  of  salt  mixed  with  ice  crystals. 

1  GUTHRIE,  Philosophical  Magazine,  4th  S.,  v.  49,  pp.  206  and  266,  1875; 
5th  S.,  v.  i,  pp.  49,  354,  and  446,  1876;  v.  2,  p.  211,  1876. 

2  PONSOT,  Annales  de  Chimie  et  de  Physique,  7th  S.,  v.  10,  p.  79,  1897. 


B1VARJANT  SYSTEMS.    TRANSITION  AND  EUTEXIA.    251 

These  arguments  have  led  Guthrie  *  to  renounce  the  hypothesis 
of  definite  cryohydrates  and  to  propose,  to  denote  these  substances, 
the  name  of  eutectic  mixtures,  which  is  generally  employed  to-day. 

210.  Eutectic  point  between  ferric  chloride  hydrates.  Inves- 
tigations of  Bakhuis  Roozboom.  —  Of  all  the  solid  hydrates  which 
a  salt  solution  may  form,  ice  is  always  the  most  hydrated  and  the 
anhydrous  salt  the  least  hydrated;  always  less  rich  in  water  than 
the  first,  the  solution  is  always  richer  in  water  than  the  second; 
the  solubility  curve  of  the  first  is  reduced  to  its  upper  branch,  the 
solubility  curve  of  the  other  to  its  lower  branch;  when  these  two 
branches  meet  their  point  of  intersection  is  necessarily  a  eutectic 
point. 

But  these  eutectic  points  are  not  the  only  ones  which  may  be 
met  with;  when  the  upper  branch  of  the  solubility  curve  of  a 
hydrate  meets  the  lower  branch  of  the  solubility  curve  of  another 
hydrate  less  rich  in  water,  the  point  of  meeting  is  a  eutectic  point. 

The  most  remarkable  example  of  such  eutectic  points  has  been 
furnished  by  Roozboom  2  from  the  study  of  the  solubility  of  ferric 
chloride. 

If  we  count  the  hydrate  of  zero  concentration  —  ice  —  and  the 
hydrate  of  infinite  concentration  —  anhydrous  ferric  chloride  —  six 
different  hydrates  of  ferric  chloride  may  be  obtained  which  are, 
in  the  order  of  increasing  concentration: 

HLO  (ice)  ; 


7H20; 
Fe4Clfl  -5H..O; 
Fe,CV  4HP; 
Fe2Cl6. 

The  first  and  last  of  these  solids  excepted,  each  of  these  sub- 
stances corresponds  to  a  solubility  curve  consisting  of  two  branches 
meeting  in  an  indifferent  point;  in  the  preceding  chapter  (p.  225  ) 
we  have  indicated  the  temperatures  to  which  these  four  indif- 
ferent points  correspond. 

1  GUTHRIE,  Philosophical  Magazine,  5th  S.,  v.  17,  p.  462,  1884. 
3  ROOZBOOM,  Archives  neerlandaises  des  Sciences  exactes  et  natureUes,  v.  27, 
1892;  Zeitschrift  fur  physikalische  Chemie,  v.  10,  p.  477,  1892. 


252 


THERMODYNAMICS  AND  CHEMISTRY. 


Fig.  74  represents-  these  various  solubility  curves;  tempera- 
tures in  centigrade  have  been  taken  as  abscissae;  as  ordinate  the 
value  N2  of  the  number  of  molecules  of  anhydrous  ferric  chloride 
contained  in  100  molecules  of  water  in  the  solution  has  been  taken. 

It  is  seen  from  this  figure  that  the  upper  branch  of  the  solu- 
bility curve  of  each  hydrate  meets  the  lower  branch  of  the  solubility 
curve  of  the  hydrate  following  immediately  in  the  order  of  in- 
creasing concentrations;  further,  thanks  to  the  phenomenon  of 

No 


20       0       20      40      60 

FIG.  74. 


100     T 


supersaturation,  by  avoiding  the  introduction  of  crystalline  par- 
ticles of  the  hydrate  Fe2Cl6-7H2O,  one  may  observe  the  inter- 
section of  the  upper  branch  of  the  solubility  curve  for  the  hydrate 
with  12  molecules  of  water  and  the  lower  branch  of  the  solubility 
curve  for  the  hydrate  with  5  molecules  of  water.  One  may,  there- 
fore, from  the  study  of  systems  formed  of  ferric  chloride  and  water, 
recognize  the  existence  of  six  eutectic  points.  Among  these  points 
there  are  five  whose  properties  are  completely  established  by  the 
researches  of  Roozboom;  these  properties  are  resumed  in  the 
following  table: 


Value  of  N2 

Eutectic 
Point. 

Hydrates  between  which  Eutexia 

is  Produced. 

Temperature 
of  Eutexia. 

for  the 
Eutectic 

Mixture. 

Wl 

Ice 

Fe2Cl 

6  12H2O 

-55°C. 

2.75 

W2 

Fe,Cl6-12H2O 

Fe2Cl 

e      7HO 

27.4 

8.23 

W3 

Fe2Cl0.12H2O 

Fe2Cl 

6     5H20 

not  studied 

W< 

Fe2Cl6-   7H2O 

Fe2Cl 

6     5H,0 

30 

6.66 

^5 

Fe2CL.   5H2O 

Fe2Cl 

6    4H20 

55 

20.32 

™« 

Fe2Cl6.  4H20 

Fe2Cl 

6 

66 

29.20 

BIVARIANT  SYSTEMS.     TRANSITION  AND  EUTEXIA.  253 

In  the  example  we  have  just  cited,  eutectic  points  alone  are 
met  with;  we  shall  next  indicate  some  remarkable  examples  where 
both  eutecti?  points  and  transition-points  are  encountered. 

2ioa.  Hydrates  of  perchloric  acid.  Van  Wyk's  investiga- 
tions.— An  example  very  similar  to  the  preceding  is  furnished 
by  the  systems  composed  of  perchloric  acid  and  water.1 

In  the  presence  of  liquid  mixtures  of  perchloric  acid  and  water 
six  kinds  of  crystals  may  be  observed,  whose  respective  composi- 
tions are: 

HCKV  H20; 
HC1O4-2H2O; 
HC1O4-3H2O; 
HC104-4H2O; 
HC1O4-6H2O; 
H20. 

If  centigrade  temperatures  are  taken  for  abscissae,  and  for  ordi- 
nates  the  ratio  of  the  number  AT2  of  molecules  of  perchloric  acid, 
HC1O4,  to  the  sum  (.A^  +  JV,)  of  the  numbers  of  molecules  of  water 
and  perchloric  acid  within  the  liquid  mixture,  the  solubility  curves 
have  the  following  appearance  (Fig.  A). 


Except  the  fusion  curve  of  ice,  which  rises  from  right  to  left, 
each  of  the  solubility  curves  is  formed  of  two  branches  meeting 
in  an  indifferent  point;  one  observes,  therefore,  in  all,  five  indif- 
ferent points:  C,  E,  G,  I,  L.  The  upper  branch  of  the  solubility 
curve  of  each  hydrate  meets  the  lower  branch  of  the  solubility 
curve  of  the  hydrate  which  immediately  precedes  it  in  the  above 

1  VAN  WYK,  Zeitschrift  fur  anorganische  Chemie,  v.  32,  p.  115,  1902. 


254 


THERMODYNAMICS  AND  CHEMISTRY. 


table;  there   are   thus   obtained  five   eutectic   points:    B,  D,  F, 
H,  K. 

211.  Researches  of  Van't  Hoff  and  Meyerhoffer  on  magne- 
sium chloride. — Van't  Hoff  and  Meyerhoffer1  have  made  a  thor- 
ough study  of  the  solubility  of  several  hydrates  of  chlorine  and  of 
magnesium.  Including  ice,  the  hydrates  furnished  by  magne- 
sium chloride  are  six  in  number: 

Ice; 

MgCl2.12H20; 

MgCl2-8H26; 

MgCl2-6H20; 

MgCl2-4H2O; 

MgCl2-2H2O. 

Further,  the  hydrate  MgCl2-8H2O  exists  in  two  different  forms 
which  we  shall  denote  by  the  symbols  a  and  /?. 

The  solubility  curves  of  these  various  hydrates  are  arranged 
as  shown  in  Fig.  75,  the  ordinates  representing  the  number  N2  of 
molecules  contained  in  100  molecules  of  water. 


20 


10 


•8a 


50° 


100° 


150° 


FIG.  75. 

In  their  investigations,  Van't  Hoff  and  Meyerhoffer,  instead  of 
the  concentration  s  of  the  solution,  make  use  of  the  number  y 

1  VAN'T  HOFF  and  MEYERHOFFER,  Sitzungsberichte  der  Berliner  Akad., 
Feb.  4  and  Feb.  18,  1897;  Zeitschrift  fur  physikalische  Chemie,  v.  27,  p.  75, 
1898. 


BIVARIANT  SYSTEMS.     TRANSITION  AND  EUTEXIA.  255 

which  put  into  the  formula  MgCl2-?/H2O  would  represent  the  con- 
stitution of  the  solution. 

They  have  especially  noted: 

1°.  The  fusing-point  of  pure  ice  (T=0°,  y=  oc); 

2°.  The  point  of  intersection  of  the  congelation  curve  and  of 
the  solubility  curve  of  the  hydrate  MgCl2  •  12H2O ;  this  is  a  eutectic 
point  for  which 

77=-33°.6,  */=20.3; 

3°.  The  point  C,  indifferent  point  for  MgCl2-12H2O: 
77=-16°.3,  2/=12.0; 

4°.  The  point  D,  intersection  of  the  solubility  curves  for 
MgCl2-12H20  and  MgCl2-8H2Oa;  it  is  a  eutectic  point  for  which 

r=-16°.7,  y=  11.17; 

5°.  The  point  E,  intersection  of  the  solubility  curves  of 
MgCl2  •  8H2Oa  and  MgCl2  •  6H2O ;  this  is  a  transition-point  for  which 

T=-3°A,  2/=10.0; 

6°.  The  point  F,  intersection  of  the  solubility  curves  of 
MgCl2  •  6H20  and  MgCl2  •  4H2O ;  it  is  a  transition-point  for  which 

T^lie^CT,  y=6.18; 

7°.  The  point  G,  intersection  of  the  solubility  curves  of 
MgCl2  •  42O  and  Mg2Cl  •  2H2O ;  this  is  a  transition-point  for  which 

77=181°.5,  2/=4.2. 

They  have,  besides,  studied  a  certain  number  of  branches  which 
the  phenomena  of  supersaturation  also  render  observable;  among 
these  branches,  which  are  shown  by  dotted  lines  in  Fig.  75,  is  the 
solubility  curve  D'E'  of  the  hydrate  MgCl2  •  8H2O/?.  The  eutectic 
point  D'  between  MgCl2-12H,O  and  MgCl2-8H2O^  corresponds  to 

T=-17°A,  2/=H.l, 

while.the  transition-point  F'  between  MgCl2  •  8H20£  and  MgCl2  •  6H2O 
has  the  coordinates 

77=-9°.6,  y 


256 


THERMODYNAMICS  AND  CHEMISTRY. 


The  phenomena  of  supersaturation  also  permitted  the  observa- 
tion of  the  point  H,  eutectic  point  between  ice  and  the  hydrate 
MgCl2-8H2Oa,  for  which 

r=-500,  2/=16.9, 

as  well  as  the    point  I,  eutectic    point    between    the    hydrates 
MgCl2-12H2O  and  MgCl2-6H2O,  for  which 

T=-19°.4,  2/=10.6. 

212.  Roozboom's  researches  on  calcium  chloride. — Another 
remarkable  example  is  given  by  the  hydrates  of  calcium  chloride, 
objects  of  an  important  memoir  by  Roozboom.1 

Including  ice,  there  are  six  hydrates  of  calcium  chloride: 

Ice; 

CaCl2-6H20; 

CaCl2-4H2Oo:; 

CaCl2-4H2O/?; 

CaCl2-2H20; 

CaCl2-  H2O. 

The  solubility  curves  of  these  hydrates  are  shown  in  Fig.  76; 


D'D 


& 


O  T 

FIG.  76. 

in  this  figure,  where  the  scale  is  not  given,  the  dotted  lines  repre- 
sent the  equilibrium  states  which  may  be  observed  for  the  cases 
of  supersaturation. 


1  ROOZBOOM,  Recueil  des  Travaux  chimiques   des  Pays  Bos,  v.  8,  p.  4 ; 
Archives  neerlandaises  des  Sciences  exactes  et  naturelks,  v.  13,  p.  199. 


BIVARIANT  SYSTEMS.    TRANSITION  AND  EUTEXIA.  257 

The  following  points  on  this  figure  are  to  be  noted; 
1°.  The  point  A,  fusing-point  of  pure  ice: 

!T=0,  s=0; 

2°.  The  point  B,  in  ersection  of  the  congelation  curve  and  the 
solubility  curve  of  CaCl2-6H2O;  this  is  a  eutectic  point  where 

r=-55°,  5=0.425; 

3°.  The  point  I,  indifferent  point  of  CaCl2-6H20: 
!F=+300.2,  s=1.027; 

4°.  The    point   C,    intersection    of    the    solubility   curves    of 
CaClj-GH^O  and  CaCl2-4H2Oa;  this  is  a  transition-point  for  which 

77=+29°.8,  s=1.006; 

5°.  The  point  C",  intersection  of  the  curves  of  solubility  for 
CaCV6H20  and  CaCl2-4H2O/?;   a  eutectic  point  : 

T=+29°.2,  s=1.128; 

6°.  The  point   D,   intersection    of    the    solubility  curves    of 
CaCl2-4H2Oa  and  CaCl2-2H2O;  this  is  a  transition-point  for  which 

77=+45°.3,  s=1.302; 

7°.  The  point  D'.   intersection  of  the  solbiulity  curves     of 
CaCl2-4H20£  and  CaCl2-2H2O;  it  is  a  transition-point: 

T=+38°.4,  s=1.275; 

8°.  The    point  E    intersection    of    the    solubility  curves    of 
CaCl2-2H,0  and  CaCl^H20;  a  transition-point  : 

!T=+174°,  5=2.757. 


258 


THERMODYNAMICS  AND  CHEMISTRY. 


The  indifferent  point  of  CaCl2-2H2O  would  be  very  close  to 
this  point,  for  it  would  correspond  to 

77=+176°     and     s=3.08, 

but  this  point  would  not  be  observed. 

It  is  probable  that  in  the  neighborhood  of  260°  there  would 
be  found  the  intersection  of  the  solubility  curves  of  CaCl  2  •  H2O 
and  of  CaCL,  which  would  be  a  new  transition-point. 

213.  Stortenbeker's  studies  on  iodine  chlorides. — A  mixture 
of  iodine  and  of  chlorine  in  the  liquid  state,  studied  with  great 
care  by  Siortenbeker,1  is  comparable  in  all  respects  to  the  mixture 
formed  by  water  and  an  anhydrous  salt ;  the  solids  that  this  mixture 
will  deposit  in  the  circumstances  in  which  Stortenbeker  was  work- 
ing are  the  substances 

I, 

ICla, 

Idft 

IC13. 

Lay  off  temperatures  as  abscissae,  and  as  ordinates  the  number 
y  which,  put  into  the  formula  Ir/Cl,  gives  the  composition  of  the 
solution;  each  of  the  four  substances  we  have  mentioned  corre- 
sponds to  a  solubility  curve;  these  four  curves  are  as  shown  in 
Fig.  77;  in  this  figure  the  dotted  lines 
can  be  observed  only  on  account  of  super- 
saturation  phenomena. 

The  following  points  are  to  be  noted 
in  this  figure: 

1°.    The    point    A,    fusing-point    of 
iodine  i' 


I  r  i  i  i  i  i 


-  C 


2°.  The  point  B,  intersection  of  the 
FIG.  77.  curve  of  congelation  for  iodine  and  the 


1  W.  STORTENBEKER,  Recueil  des  Travaux  chimiques  des  Pays  Bas,  v. 
1888;  Zeitschrift  fur  physikalische  Chemie,  v.  3,  p.  11,  1888. 


BIVARIANT  SYSTEMS.    TRANSITION  AND  EUTEXIA.  259 
solubility  curve  -of  ICla;  this  is  a  eutectic  point  for  which 
r=+7°.9,  2/=0.66; 

3°.  The  point  E'  of  intersection  of  the  congelation  curve  for 
iodine  and  the  solubility  curve  of  IC1/?;  also  a  eutectic  point  whose 
coordinates  are  not  well  known; 

4°.  The  indifferent  point  7  for  Ida;  the  coordinates  of  this 
point  are 

T=27°.2,  2/=l; 

5°.  The  indifferent  point  /'  of  the  chloride  IC1/?,  of  coordinates 
!T=130.9;  2/=l; 

6°.  The  point  of  intersection  C  of  the  solubility  curve  of  the 
monochloride  ICla  and  of  the  solubility  of  the  trichloride  IC13; 
this  is  a  eutectic  point  for  which 

r=22°.7,  2/=1.19; 

7°.  The  indifferent  point  J  of  the  trichloride  ICL,;  the  coordi- 
nates of  this  point  are 


214.  Studies  by  Guthrie  and  by  Le  Chatelief  on  the  mixtures 
of  two  salts.  —  Instead  of  observing  the  eutectic  mixtures  of  ice 
and  salt  obtained  by  cooling  an  aqueous  solution  of  this  salt,  one 
may  cool  the  liquid  obtained  by  dissolving  a  salt  in  another  melted 
salt;  in  this  way  are  obtained  eutectic  mixtures  of  the  two  salts. 

In  this  case  it  is  convenient  to  represent  the  composition  of  the 

liquid,  not  by  the  ratio  s=  ^  of  the  mass  M2  of  the  salt  2  to  the 

Ml 

mass  Mj  of  the  salt  1,  but  by  the  ratio  x=  M     l^-  of  the  mass 

of  the  salt  2  to  the  total  mass  (M^-\-M^  of  the  liquid;  this  ratio 
may  vary  from  the  value  z=0,  which  corresponds  to  the  salt  1 
in  the  pure  state,  to  the  value  x=l,  which  represents  the  salt  2 
taken  in  the  pure  state. 


260 


THERMODYNAMICS  AND   CHEMISTRY. 


F2 


Let  us  take  the  values  of  T  for  abscissae  and  the  values  of  x  for 
ordinates.  The  congelation  curve  of  the  salt  1  in  the  mixture  will 
be  a  curve  C^  (Fig.  78) ;  this  curve  will  start  from  the  point  Fl  of 

abscissa  TL,  which  is  the  fusing-point  of 
the  salt  1  in  the  pure  state  and  whose 
ordinate  is  x  =  0;  it  will  rise  from  right 
to  left.  The  curve  of  congelation  of 
the  salt  2  will  be  a  curve  C3  which, 
starting  from  F2  of  abscissa  T2,  the 
fusing-point  of  the  pure  salt  2  whose 
ordinate  is  x  =  l,  will  descend  from 
right  to  left. 

Tl       Tz     T  These  two  curves  will  intersect  at 

FlG-  78-  the  eutectic  point  a>,  of  abscissa  6  and 

ordinate  £. 

Guthrie  l  has  studied  a  number  of  eutectic  mixtures  formed 
by  melted  salts;  below  are  the  coordinates  6,  £  of  the  eutectic 
points  of  some  of  these  mixtures: 


Mixed  Salts. 

e. 

£. 

1.  Potassium  nitrate              ) 

2.  Lead  nitrate                       \ 

207° 

53.14 

1.  Potassium  nitrate              1 

2.  Calcium  nitrate                  l 

251° 

74.64 

1.  Potassium  nitrate 

2.  Strontium  nitrate 

258° 

74.19 

1.  Potassium  nitrate 

2.  Barium  nitrate 

278° 

70.47 

1.  Potassium  nitrate             ) 

2.  Potassium  chromate.  .  .  .  f 
1.  Potassium  nitrate  | 
2.  Potassium  sulphate.  .  .  .  j 
1.  Potassium  nitrate  / 
2.  Sodium  nitrate     .           .  ) 

295° 
300° 
215° 

96.24 
97.64 
67.10 

1.  Sodium  nitrate  ) 
2.  Lead  nitrate  f 

268° 

57.16 

Guthrie  has  also  traced  the  curves  C19  C2  for  the  mixture  of 
potassium  nitrate  and  lead  nitrate. 

Le  Chatelier2  has  likewise  studied  the  phenomena  of  eutexia 

1  GUTHRIE,  Philosophical  Magazine,  5th  S.,  v.  17,  p.  462,  1884. 
*LE  CHATELIER,  Comptes  Rendus,  v.  118,  p.  709,  1894. 


BIVARIANT  SYSTEMS.     TRANSITION  AND  EUTEXIA.  261 


which  are  observed  by  cooling  the  mixture  of  two  melted  salts. 

With  melted  sodium  chloride  he  has  mixed  sodium  carbonate,  the 

neutral  pyrophosphate  of  sodium  and  barium  x 

chloride;  with  melted  lithium  sulphate  he  has 

mixed  calcium  sulphate,  sodium  sulphate  and 

lithium  carbonate ;  for  all  these  cases  he  has 

found  the  phenomena  of  eutexia. 

The  phenomenon  becomes  complicated 
when  the  two  melted  mixed  salts  may  form  a 
double  salt ; 1  to  the  two  curves  C,  and  C2  it  is 
necessary  in  this  case  to  join  the  solubility  curve 
D  (Fig.  79)  of  the  double  salt  in  the  liquid  mix- 
ture; this  curve  possesses  in  general  an  indif- 
ferent point  7;  its  points  of  intersection  atlt  aj2 
with  the  curve  Clt  C,  are  eutectic  points. 


FIG.  79. 


1  LE  CHATELIER,  Comptes  Rendus,  v.  113,  p.  801,  1894. 


CHAPTER  XIII. 
MIXED   CRYSTALS.    ISOMORPHOUS   MIXTURES. 

215.  Isomorphous  salts;  Riidorff's  observations. — Let  us  take 
two  salts  incapable  of  reacting  chemically,  for  example,  ammo- 
nium chloride  and  ammonium  nitrate ;  put  them  in  contact  with 
a  quantity  of  water  too  small  to  entirely  dissolve  either  one  of  the 
two  salts;  three  independent  components,  water,  ammonium 
chloride,  ammonium  nitrate  make  up  the  system;  this  system  is 
besides  divided  into  three  phases,  the  two  crystallized  salts  and 
the  aqueous  solution  of  these  two  salts;  we  have  therefore  to  deal 
with  a  bivariant  system  which  may  be  put  in  equilibrium  under 
every  pressure  and  at  all  temperatures,  when  equilibrium  is  estab- 
lished at  a  given  pressure  and  temperature;  the  solution,  saturated 
with  each  of  the  two  salts,  should  have  a  perfectly  definite  com- 
position, independent  of  the  masses  of  ammonium  chloride,  of 
ammonium  nitrate  and  water  that  coexist  with  this  solution. 

This  is  what  experiments,  made  a  good  while  ago,  revealed 
to  Rudorff,1  not  only  for  what  concerns  the  two  salts  o  which 
we  have  spoken,  but  also  for  a  certain  number  of  pairs  of  salts 
incapable  of  any  chemical  reaction,  either  because  they  are  from 
the  same  base  or  from  the  same  acid. 

But  Rudorff  also  found  a  certain  number  of  saline  couples 
which  do  not  obey  the  rule  above  stated. 

For  instance,  in  place  of  ammonium  chloride  and  ammonium 
nitrate,  take  potassium  sulphate  and  ammonium  sulphate;  put 
them  in  the  presence  of  a  quantity  of  water  incapable  of  dissolving 
them  totally;  at  a  given  temperature  and  pressure  the  system 

*  RUDORFF,  Poggendorf's  Annalen,  v.  148,  p.  456,  1873. 

262 


MIXED  CRYSTALS.    1SOMORPHOUS  MIXTURES.       263 

still  comes  to  an  equilibrium  condition;  but  the  composition  of 
the  solution,  for  the  system  in  equilibrium,  is  no  longer  determined 
by  the  knowledge  of  the  pressure  which  the  system  supports  and 
of  the  temperature  to  which  it  is  brought;  it  still  depends  upon 
the  relative  values  of  the  masses  of  the  two  alts  and  water  which 
have  been  put  together;  if,  without  changing  pressure  or  tem- 
perature, there  is  added  a  certain  quantity  of  one  or  the  other  of 
the  two  salts,  the  solution  changes  in  composition,  enriches  itself 
relatively  to  the  salt  added,  and  becomes  poorer  in  the  other  salt. 

216.  Interpretation  of  the  preceding  facts;  isomorphous  mix- 
tures are  solid  solutions. — These  properties  do  not  belong  to  a 
bi variant  system;  a  multi variant  system  alone  can  possess  them; 
it  follows,  therefore,  that  the  calcu  ation  which  has  caused  us  to 
consider  the  system  bivariant — ammonium  chloride,  ammonium 
nitrate,  water — is  false  in  some  way  when  we  try  to  extend  it 
to  the  system  potassium  sulphate  ammonium  sulphate,  water. 
Now  the  error  evidently  cannot  be  in  the  number  of  independent 
components,  a  number  certainly  equal  to  3;  it  can  therefore  only 
be  in  the  number  of  phases;  the  number  of  phases  into  which  the 
system  is  divided  when  equilibrium  is  reached  cannot  be  equal 
to  3;  it  cannot  exceed  2.  Whence  comes  this  redu  tion  in  the 
number  of  phases? 

Potassium  sulphate  and  ammonium  sulphate  are  two  isomor- 
phous salts ;  when  masses  of  these  two  salts  are  left  a  long  time  in 
contact  with  an  aqueous  solution  the  crystals  of  both  cease  to  be 
distinct,  and  at  last  there  rest  only  mixed  crystals,  containing  both 
sulphate  of  potassium  and  ammonium  sulphate. 

Riidorff's  experiments,  compared  with  the  theorems  of  J.  Wil- 
lard  Gibbs,  show  us  that  the  mixed  crystals  should  be  considered 
not  as  two  phases,  but  as  a  single  phase  these  crystals  are  not 
therefore,  as  many  writers  have  supposed  simp  y  mechanical 
mixtures,  a  juxtaposition  or  a  mixing  of  crystalline  particles  of 
potassium  sulphate  and  crystalline  pa  'tides  of  ammonium  sulphate ; 
in  them  the  two  component  salts  are  physicaUy  mixed  in  a  manner 
as  intimate  as  for  an  aqueous  solution;  every  volume,  however 
small  which  may  be  cut  from  one  of  these  crystals,  contains  a 
certain  quantity  of  each  one  of  these  salts;  these  mixed  crystals, 
formed  by  two  isomorphous  bodies,  constitute,  according  to  the 


2p4  THERMODYNAMICS  AND  CHEMISTRY. 

expression  created  by  Van't  Hoff  in  considering  other  facts,  a  solid 
solution. 

217.  Theory  of  the  solubility  of  two  isomorphous  salts. — This 
assimilation  into  a  solid  solution  of  mixed  crystals  formed  by  two 
isomorphous  salts  leads  to  a  complete  theory  of  the  phenomena 
which  are  produced  when  two  isomorphous  salts  are  brought  into 
the  presence  of  water. 

We  have  in  fact  here  a  system  formed  of  three  independent  com- 
ponents water  0  and  the  two  salts  1  and  2 ;  this  system  is  divided 
into  two  phases,  the  liquid  solution  for  which  we  shall  continue 
to  indicate  by  slr  S2,  the  two  concentrations,  and  the  mixed  crystals 
C;  this  system  is  therefore  invariant;  when  the  temperature  and 
pressure  only  are  given,  the  composition  of  each  of  the  two  phases 
capable  of  remaining  in  equilibrium  in  contact  with  each  other  is 
not  completely  determined;  it  becomes  entirely  determined  if  to 
the  temperature  and  pressure  there  is  added  another  given  quan- 
tity, for  example,  one  of  the  concentrations  st  of  the  solution. 

Suppose  the  pressure  TT  given  once  for  all  and  equal,  for  instance, 
to  the  atmospheric  pressure.  Whenever  there  is  given  the  tem- 
perature T  and  the  first  concentration  st  of  the  liquid  solution, 
the  econd  concentration,  s2,  should  have  a  well-determined  value 
if  the  liquid  solution  is  to  remain  in  equilibrium  in  contact  with 
mixed  crystals;  if,  therefore,  as  in  Art.  102,  we  lay  off  on  the  three 
coo  dinate  rectangular  axes  the  values  of  the  temperature  T  and 
of  the  concentrations  st  and  s2,  we  shall  find  that  the  temperature 
and  the  concentrations  of  every  solution  capable  of  remaining 
in  equilibrium  in  contact  with  mixed  crystals  C  are  the  coordinates 
of  a  point  M  situated  on  a  surface  S,  a  conclusion  similar  to  that 
reached  for  the  case  in  which  the  solid  C  was  a  chemical  com- 
pound o  definite  composition. 

But,  and  in  this  the  problem  we  are  treating  is  more  complicated 
than  that  of  Art.  102,  the  solution  whose  properties  (temperature 
and  concentrations)  are  represented  by  the  coordinates  of  a  point 
M  of  the  surface  S  does  not  remain  in  equilibrium  with  any  mixed 
crystals  whatever;  the  mixed  crystals  which  may  remain  in  equi- 
librium in  contact  with  this  solution  have  a  well-determined  com- 
position, which  varies  as  the  point  M  assumes  successively  dif- 
ferent positions  on  the  surface  S. 


MIXED  CRYSTALS.    ISOMORPHOUS  MIXTURES.       265 

These  principles,  necessary  consequence  of  Gibbs'  theories, 
have  been  brought  out  for  the  most  part  by  Roozboom  1  and  his 
pupils. 

218.  Isomorphism  of  the  sulphates  of  the  magnesium  series. 
Studies  of  Stortenbeker. — The  most  complete  experimental  re- 
searches which  have  been  made  on  this  subject  are  due  to  Stor- 
tenbeker.2 They  deal  with  the  phenomena  of  isomorphism,  pos- 
sessed by  the  different  hydrates  of  sulphates  of  the  magnesium 
series: 

MgSO4,    ZnS04,    FeS04,    CuS04,    MnSO4,    CdSO4. 

These  cases  of  isomorphism  had  already  been  studied  by  Mit- 
scherlich.  When  two  of  these  sulphates  are  dissolved  in  water,  the 
solution  may  precipitate  mixed  crystals ;  but  in  general,  according 
to  the  temperature  and  composition  of  the  solution,  one  may 
obtain  various  kinds  of  mixed  crystals. 

Let  us  take,  for  example,  the  case  so  well  studied  by  Storten- 
beker 2  where  the  solution  contains  zinc  sulphate  and  copper  sul- 
phate; three  kinds  of  mixed  crystals  may  be  obtained,  namely: 

Triclinic  crystals  (anorthic)  corresponding  to  the  formula 
(Zn,Cu)SO4-5H2O; 

Monoclinic  crystals  (clinorhombic)  of  formula  (Zn,Cu)SO4-7H20; 

Orthorhombic  crystals  having  the  same  formula. 

Concerning  the  so  ubility  surfaces  of  these  crystals  one  may 
repeat  all  that  has  been  said  in  Arts.  103,  104,  and  105  about  the 
surfaces  of  solubility  of  double  salts. 

To  each  of  the  mixed  crystals  corresponds  a  solubility  surface 
referred  to  the  axes  OT,  Osl}  Os2;  but  certain  parts  of  this  surface 
will  represent,  in  general,  states  of  equilibrium  observable  only  in 
solutions  supersaturated  with  respect  to  another  kind  of  crystal; 
if  these  parts  are  suppressed  so  as  to  keep  only  the  repre- 
sentation of  equilibrium  states  where  all  supersaturation  is  ex- 
cluded, a  polyhedron  will  be  obtained  with  curved  faces  having 
as  many  faces  as  there  are  kinds  of  mixed  crystals.  The  edges 

1  ROOZBOOM,  Archives  neerlandaises  des  Sciences  exactes  et  naturelles,  v.  26, 
p.  137,  1891;  Zeitechrift  fur  physikalische  Chemie,  v.  8,  p.  504,  1891. 

2  STORTENBEKER,  Zeitschrift  fur  physikalische  Chemie,  v.  22,  p.  60,  1897. 


266 


THERMODYNAMICS  AND  CHEMISTRY. 


of  this  polyhedron  will  represent  the  solutions  which  may  rest 
in  equilibrium  in  contact  with  two  distinct  kinds  of  mixed 
crystals. 

For  the  mixtures  of  copper  sulphate  and  zinc  sulphate  Storten- 
beker  has  not  constructed  the  entire  surface  of  which  we  have  just 
spoken,  but  only  the  points  on  this  surface  which  correspond  to 
the  temperature  T  =  18°.  In  the  system  of  axes  Osl}  Os2  (Fig.  80), 

where  st  represents  the  concen- 
tration in  copper  sulphate  and  s2 
the  concentration  in  zinc  sulphate, 
he  obtains  three  lines  which  corre- 
spond to  the  three  kinds  of  mixed 
crystals.  The  dotted  portions  of 
these  lines  can  only  be  observed 
thanks  to  the  phenomena  of  super- 
saturation. 

It  is  seen  that  at  18°  the  tri- 
clinic  crystals  with  5  molecules  of 
water  are  obtained  as  long  as  the 
amount  of  zinc  sulphate  does  not 
exceed  a  certain  limit;  there  are 


si 


A 

FIG.  80. 

next  obtained  clinorhombic  crystals  of  7  molecules  of  water;  finally, 
the  orthorhombic  crystals  of  7  molecules  of  water  are  deposited  on  y 
from  solutions  very  poor  in  copper  sulphate. 

The  information  given  by  this  figure  does  not  suffice  to  make 
known  all  the  properties  possessed  at  18°  by  isomorphous  mixtures 
of  copper  sulphate  and  zinc  sulphate;  it  is  further  necessary  to 
know  the  proportion  of  zinc  and  of  copper  in  the  mixed  crystals 
deposited  from  the  solution  represented  by  each  of  the  points  of 
the  various  lines  shown  in  Fig.  80.  Stortenbeker  has  made  known 
this  proportion;  he  has  constructed  curves  which  determine,  for 
the  temperature  18°,  the  composition  of  the  crystals  when  one 
knows  the  composition  of  the  solution  which  rests  in  equilibrium 
in  contact  with  these  crystals. 

Take  a  number  of  grammes  of  each  crystal  equal  to  its  molecu- 
lar weight  and  determine  the  number  n  of  copper  atoms  which  are 
therein  contained.  When  we  follow  the  line  AB  from  A  to  B 
within  the  triclinic  crystals  with  5  water  molecules,  n  varies  from 


MIXED  CRYSTALS.     ISOMORPHOUS  MIXTURES.        267 


1  to  0.828;  when  we  follow  the  line  BC  from  B  to  C  in  the  midst  of 
the  clinorhombic  crystals  with  7  water  molecules,  n  changes  from 
0.319  to  0.149;  finally,  when  we  follow  the  line  CD  from  C  to  D 
within  the  orthorhombic  crystals  with  7  molecules,  n  varies  from 
0.0197  to  0. 

Therefore,  at  a  given  temperature,  if  all  phenomena  of  super- 
saturation  are  excluded,  the  crystals  of  each  kind  which  can  be 
obtained  have  a  composition  which  remains  included  between  two 
given  limits;  between  the  limiting  compositions  of  the  crystals  of 
two  different  kinds  there  exist  gaps;  certain  compositions  corre- 
spond to  no  kind  of  crystal  susceptible  of  remaining  in  equilibrium, 
at  the  temperature  considered,  with  a  solution  freed  from  all  super- 
saturation. 

The  phenomena  which  we  have  just  described  and  the  curves 
which  represent  them  change  with  the  temperature.  Stortenbeker 
has  not  fol  owed,  for  the  above  ca  e,  this  influence  of  temperature; 
out  he  has  examined  it  in  studying  the  isomorphous  mixtures  of 
magnesium  sulphate  and  of  copper  sulphate.1 

There  are  here  two  kinds  of  mixed  crystals :  tri-clinic  (anorthic) 
corresponding  to  the  formula  (Cu,MnS04  •  5H2O  and  clinorhombic 
crystals  of  formula  (Cu,Mn)S04-7H2O. 

Take  sl  for  the  concentration  of  copper  sulphate  and  s2  for  that 
of  manganese  sulphate   and  at  each  tem- 
perature draw  a  solubility  curve  for  the 
two  kinds  of  mixed  crystals  referred  to 
the  axes  stOs2. 

At  18°  the  solubility  curves  are  ar- 
ranged as  indicated  in  Fig.  81 ;  in  this 
figure  the  dotted  lines  represent  solu- 
tions saturated  with  respect  to  one  kind 
of  c  ystal,  but  supersaturated  with  re- 
spect to  the  other. 

If  one  supposes  excluded  all  kinds  of 
supersaturation,  it  is  seen  that  the  tri- 
clinic  crystals  with  5  water  molecules  are 
those  obtained  either  in  contact  with  solutions  rich  in  copper, 


1  STORTENBEKER,  Zeitschrift  fur  physikalische  Chemie,  v.  34,  p.  Ill,  1900. 


268 


THERMODYNAMICS  AND  CHEMISTRY. 


or  in  contact  with  solutions  rich  in  manganese;  only  the  solu- 
tions of  an  intermediate  composition  can  furnish  clinorhombic 
crystals  with  7  molecules  of  water. 

If  the  number  n  is  defined  as  in  the  preceding  case,  we  see  that 
for  the  triclinic  crystals  obtained  in  these  conditions  n  is  included 
between  1  and  0  229  (corresponding  to  the  solubility  curve  AB), 
or  between  0.105  and  0  (corresponding  to  the  solubility  curve  CD) ; 
while  for  the  clinorhombic  crystals  for  which  BC  is  the  solubility 
curve,  n  is  included  between  0.235  and  0.16. 

At  10°  the  arrangement  of  the  solubility  curves  is  that  repre- 
sented by  Fig.  82;  the  solutions  rich  in  copper  continue  to  give 
triclinic  crystals  with  5  molecules  of  water,  but  the  solutions  rich 
in  manganese  give  clinorhombic  crystals  with  7  molecules  of  water. 


\ 


FIG.  82. 


FIG.  83. 


Again,  at  23°  the  solubility  curves  are  arranged  as  indicated 
in  Fig.  83.  Avoiding  all  supersaturation,  a  solution  can  remain 
in  equilibrium  only  in  contact  with  triclinic  crystals  with  5  water 
molecules  the  solutions  saturated  with  respect  to  clinorhombic 
crystals  with  7  water  molecules  are  supersaturated  with  respect 
to  the  preceding  crystals. 

From  these  data  it  is  not  difficult  to  recognize  the  general 
arrangement  of  the  solubility  surface,  limited  by  the  plane  sftSz 
(Fig.  84),  which  corresponds  to  T=23°;  the  line  ABCD,  in'er- 
section  of  the  surface  by  the  plane  T  — 18°,  is  that  represented 
by  Fig.  81. 


MIXED  CRYSTALS.    ISOMORPHOUS  MIXTURES.        269 


Stortenbeker l  has  also  studied  the  mixed  crystals  formed  by  cad- 
mium sulphate  and  ferrous  sulphate ; 
by  zinc  sulphate  and  magnesium 
sulphate;  by  magnesium  sulphate 
and  ferrous  sulphate;  by  copper 
sulphate  and  manganese  sulphate; 
by  cobalt  chloride  and  manganese 
chloride.  We  shall  limit  ourselves 
to  referring  the  reader  to  these 
valuable  memoirs. 

219.  Solutions  containing  mixed 
crystals  and  definite  compounds. 
Researches  of  Roozboom  and  of 
Retgers. — It  may  happen  that  a 
solution  of  two  salts  in  water  precipitates,  according  to  cir- 
cumstances, either  mixed  crystals  or  a  definite  compound 
such  as  a  hydrate  or  a  double  salt.  The  solubility  surface,  referred 
as  before  to  the  axes  OT,  Os  17  Os2,  is  composed  of  several  regions; 
among  these  regions  there  are  some  which  correspond  to  a  definite 
compound,  simple  salt  or  double  salt ;  there  are  others  correspond- 
ing to  mixed  crystals;  for  each  of  the  points  of  the  latter  there  is 
a  mixed  crystal  of  given  constitution;  but  this  constitution  varies 
according  to  the  point  chosen. 

An  important  example  has  been  studied  2  by  Roozboom ;  it 
is  furnished  by  the  aqueous  solutions  of  ferric  chloride  and  of  am- 
monium chloride.  Although  these  two  salts  cannot  be  considered 
amorphous,  their  solutions,  as  had  been  previously  shown  by  Leh- 
mann,  may  furnish  mixed  crystals ;  they  may  also  give  as  precipi- 
tates definite  compounds,  namely,  the  hydrate  Fe2Cl6-12H2O  and 
the  double  salt  (NH4)2  -  FeCl5  -  H2O. 

On  the  solubility  surface  each  of  these  kinds  of  precipitate 
has  its  region ;  if  the  solubility  surface  is  cut  by  a  plane  perpendic- 
ular to  OT,  in  such  manner  as  to  obtain  an  isothermal,  this  iso- 

1  STORTENBEKER,  loc.  cit.  and  Zeitschrift  fur  physikalische  Chemie,  v.  16, 
p.  250,  1895,  and  v.  17,  p.  643,  1895. 

2  ROOZBOOM,  Archives  neerlandaises  des  Sciences  exactes  et  naturelles,  v.  27, 
p.  1,  1892;  Zeitschrift  fur  physikalische  Chemie,  v.  10,  p.  145,  1892;   MOHR, 
Zeit.  f.  phys.  Chemie,  v.  27,  p.  193,  1898. 


270  THERMODYNAMICS   AND   CHEMISTRY. 

thermal  will  be  made  up  of  three  curves  which  will  represent  the 
solutions  capable  of  remaining  in  equilibrium  either  in  contact 
with  the  ferric  hydrate  or  in  contact  with  mixed  crystals. 

At  the  temperature  of  15°,  if  the  coordinate  s1  is  taken  as  con- 
centration in  ammonia  salt,  and  s2  the  concentration  in  ferric 

chloride,  these  three  curves  are  arranged 
as  shown  in  Fig.  85.  According  to  this 
arrangement,  the  mixed  crystals  are 
precipitated  from  solutions  rich  in  am- 
monia  salt,  the  ferric  hydrate  from  solu- 
tions very  poor  in  ammonia  salt,  the 
double  salt  from  solutions  of  inter- 
mediate composition. 

_          Retgers,    whose    researches 1    have 
1     contributed    greatly    to    increase    our 
FIG.  85.  knowledge  concerning  isomorphism,  has 

shown  that  this  property  to  precipitate,  according  to  circum- 
stances, either  mixed  crystals  or  a  double  salt,  belongs  very  often 
to  solutions  of  two  isomorphous  salts. 

A  disposition  which  seems  to  be  realized  frequently  is  the  fol- 
lowing : 

The  solutions  for  which  the  amount  of  salt  2  does  not  exceed 
a  certain  limit  furnish  mixed  crystals,  isomers  of  those  given  by  the 
salt  1  in  the  pure  state;  the  solutions  for  which  the  amount  of  salt 
1  does  not  exceed  a  certain  limit  precipitate  isomorphous  crystals 
of  salt  2;  finally,  solutions  of  intermediate  composition  furnish  a 
double  salt  of  definite  composition. 

In  this  way  behave  2  the  aqueous  solutions  of  the  two  sub- 
stances 

K2S04,  Na2S04. 

The  solutions  rich  in  potassium  sulphate  give  mixed  isomor- 
phous crystals  of  potassium  sulphate;  the  solutions  rich  in  sodium 
sulphate  give  mixed  isomorphous   crystals  of  sodium  sulphate; 
finally,  intermediate  solutions  give  a  double  salt  whose  formula  is 
3K2S(VNa,S04. 

1  These  researches  are  for  the  most  part  to  be  found  in  the  Zeitschrift  filr 
physikalische  Chemie. 

1  RETGERS,  Zeit.  f.  phys.  Chemie,  v.  6,  p.  226,  1890. 


MIXED  CRYSTALS.    ISOMORPHOUS  MIXTURES.        271 

Solutions  of  calcium  carbonate  and  magnesium  carbonate  be- 
have in  the  same  way;1  one  may  obtain: 

1°.  Mixed  crystals,  isomers  of  calcite,  containing  from  0  to 
0.025  of  magnesium  carbonate; 

2°.  Mixed  crystals,  isomers  of  magnesite,  containing  from  0  to 
0.03  of  calcium  carbonate; 

3°.  A  double  salt,  dolomite,  having  the  formula 

CaCOa-MgCO,. 

Retgers  was  able,  from  considerations  of  this  nature,  to  account 
for  the  peculiarities  possessed  by  the  mineralogical  series  of  pyrox- 
ene, olivine  and  pyrite.2 

220.  Two  melted  isomorphous  salts;  case  where  there  is 
produced  a  single  kind  of  mixed  crystals. — We  have  just  studied 
the  ormation  of  mixed  crystals  in  an  aqueous  solution  containing 
two  isomorphous  salts.  By  side  of  this  generation  of  mixed  crys- 
tals in  the  wet  way.  one  may  study  their  formation  in  the  dry  way; 
one  may  melt  together  two  isomorphous  substances  and  study  the 
mixed  crystals  which  the  mixture  furnishes  on  freezing ;  the  problem 
is  thus  quite  simplified,  because  we  have  to  deal  no  longer  with 
three,  but  with  only  two  independent  components. 

Roozboom  has  given  3  a  very  complete  theoretical  study  of  the 
various  peculiarities  which  may  occur,  and  his  pupils  have  added 
to  this  study  remarkable  experimental  verifications;  we  shall 
limit  ourselves  to  sketching  some  of  the  essential  traits. 

The  most  simple  case  which  can  oc  ur  is  that  where  the  liquid 
mixture  formed  of  the  sub  tances  1  and  2  never  furnishes,  whatever 
its  composition,  other  than  a  single  kind  of  mixed  crystals;  from 
the  point  of  view  of  composition,  these  latter  may  have  all  the 
intermediate  states  between  the  crystals  of  substance  1  in  the 
pure  state  and  the  crystals  of  substance  2  in  the  pure  state. 

We  have  to  do  here  with  what  we  have  called  a  double  mixture 
(Art.  182) ;  this  double  mixture  is  very  comparable  to  that  formed 

1  RETGERS,  Zeit.  /.  phys.  Chemie.,  v.  6,  p.  227,  1890. 

*  RETGERS,  Annales  de  I'Ecole  polytechnique  de  Delft,  v.  6,  p.  186,  1891. 

3  ROOZBOOM,  Archives  neerlandaises  des  Sciences  exactes  et  naturelles, 
Series  II,  v.  3,  pp.  414,  1900;  Zeitschnft  fur  physikalische  Chemie,  v.  30,  p.  385, 
and  413,  1900. 


272 


THERMODYNAMICS  AND  CHEMISTRY. 


by  a  mixture  of  two  volatile  liquids  in  the  presence  of  a  mixed 
vapor;  in  the  discussion  of  this  last  double  mixture  it  will  suffice 
almost  always  to  replace  the  words  liquid  mixture  and  mixed  vapor, 
respectively,  by  the  words  solid  solution  and  liquid  mixture  to 
obtain  the  theory  of  the  first  double  mixture. 

Let  us  represent  the  composition  of  each  of  our  two  mixtures 
as  we  have  done  in  Art.  195 ;  1  gramme  of  solid  solution  or  liquid 
contains  X  grammes  of  substance  2  and  (1— X)  grammes  of  sub- 
stance 1;  X  is  variable  from  0  to  1;  X=0  represents  the  sub- 
stance 1  in  the  pure  state;  X=l  represents  the  substance  2  in 
the  pure  state.  Lay  off  this  value  of  X  as 
abscissa,  and  along  the  axis  of  ordinates 
lay  off  the  value  T  of  the  temperature 
(Fig.  86).  Suppose  the  pressure  constant. 
Take  a  solid  solution  of  composition 
X  and  gradually  raise  its  temperature; 
the  point  representing  the  state  of  the  sys- 
tem will  mount  along  XM  parallel  to  OT. 
As  long  as  the  temperature  remains 
less  than  a  certain  value  T  the  crystals 
will  not  undergo  any  trace  of  fusion. 

The  instant  the  temperature  reaches 
the  value  T  the  representative  point 
being  then  at  M,  there  will  appear  the 


Liquid 


O      x        X         1 

FIG.  86. 

first  liquid  drop,  which  will  not  have  the  composition  X. 

The  temperature  increasing  above  T,  the  system  which  will 
conserve  the  mean  composition  X  will  be  in  part  in  the  crystalline 
state,  partly  in  the  liquid  state ;  neither  the  crystals  nor  the  liquid 
will  have  the  composition  X. 

When  the  temperature  reaches  a  certain  limit  T',  higher  than 
T,  the  representative  point  being  then  at  ra',  the  system  will  have 
assumed  entirely  the  liquid  state;  the  liquid,  whose  composition 
will  be  assuredly  X,  will  remain  homogeneous  at  temperatures 
higher  than  Tf. 

If  X  is  made  to  vary  from  0  to  1,  the  point  M  describes  a  cer- 
tain curve  C ;  the  point  ra'  describes  a  certain  other  curve,  c,  situ- 
ated entirely  above  the  curve  C.  For  X=Q  the  two  curves  C  and 
c  start  from  the  same  point  Flt  whose  ordinate  OF^  is  the  fusing- 


MIXED  CRYSTALS.    ISOMORPHOUS  MIXTURES.        273 

point  of  the  substance  1  in  the  pure  state;  for  X=  1  the  two  curves 
C,  c  join  in  a  point  F2,  whose  ordinate  IF2  represents  the  tem- 
perature of  fusion  of  the  pure  substance  2. 

The  two  curves  C,  c  divide  the  plane  into  three  regions.  When 
the  representative  point  is  found  in  the  region  situated  below  the 
curve  C  the  system  is  in  the  state  of  homogeneous  solid ;  when  the 
representative  point  is  above  the  curve  c  the  system  is  in  a  state 
of  homogeneous  liquid;  when  the  representative  point  is  between 
C  and  c  the  system  of  mean  composition  X  is  partly  in  the  solid 
state  and  partly  in  the  liquid. 

If  we  draw  a  parallel  TmM  to  OX,  this  line  will  meet  the  curve 
C  in  a  point  M,  of  abscissa  X,  and  the  line  c  in  a  point  m  of  abscissa 
x ;  x  represents  the  composition  of  the  liquid,  which  at  the  tempera- 
ture T  can  remain  in  equilibrium  in  contact  with  crystals  of 
composition  X. 

According  to  the  opinion  pretty  generally  held  among  the 
chemists  who  have  insufficiently  meditated  upon  the  laws  of  chemi- 
cal statics,  the  two  lines  C,  c  would  coincide  for  a  great  number 
of  cases  and  would  be  reduced  to  a  straight  line  joining  the  points 
F^  and  F2.  At  a  given  temperature  a  fluid  mixture  of  given  com- 
position would  furnish  crystals  of  the  same  composition. 

G.  Bruni *  has  very  well  shown  this  opinion  to  be  inadmissible. 
We  may,  in  fact,  apply  to  the  systems  we  are  studying  the  theo- 
rems of  Gibbs  and  of  Konovalow  (Art.  194),  and  particularly  the 
first.  It  suffices  to  substitute  for  the  words  mixed  liquid,  mixed 
vapor,  the  words  mixed  crystals,  mixed  liquid. 

If,  at  a  certain  temperature,  the  mixed  crystals  may  remain  in 
equilibrium  in  contact  with  a  liquid  mixture  of  the  same  compo- 
sition, at  this  temperature  the  two  curves  C  and  c  have  a  common 
point;  they  ought  also,  in  virtue  of  the  theroem  indicated,  to  have 
a  common  tangent  parallel  to  OX.  If,  therefore,  the  composition 
of  the  liquid  which  may  remain  in  equilibrium  in  contact  with 
mixed  crystals  is  always  identical  with  that  of  these  crystals, 
not  only  the  two  curves  C,  c  would  coalesce,  but  their  common 
tangent  would  be  constantly  parallel  to  OX;  the  two  curves 
would  therefore  be  reduced  to  a  same  straight  line  parallel  to 

1  G.  BRUNI,  Rendiconti  dell'  Accademia  dei  Lincei,  v.  7,  pp.  138  and  347, 
1898. 


274 


THERMODYNAMICS  AND  CHEMISTRY. 


OX.  That  this  may  be  possible,  it  would  be  necessary  that  the 
two  substances  1  and  2  had  the  same  fusing-point,  and  that 
the  same  be  true  of  all  the  mixed  crystals  which  they  may  produce. 
We  shall  study  an  example  of  this  last  case  in  Art.  230. 

This  does  not  mean  to  say  that  the  two  curves  C  and  c  may 
not  have,  in  certain  cases,  a  common  point  /;  at  the  temperature 
d,  which  serves  as  abscissa  to  the  point  /,  the  mixed  crystals  may 
remain  in  equilibrium  in  contact  with  a  liquid  mixture  of  the 
same  composition,  so  that  this  equilibrium  state  is  indifferent. 
At  the  indifferent  point  7  the  two  curves  have  a  common  tangent 
parallel  to  OX;  this  point  is  therefore,  for  the  two  curves,  a  point 
of  maximum  ordinate  or  of  minimum  ordinate. 

A  very  good  example  of  this  last  case  is  given  us  by  the 
mixtures  of  mercury  bromide  and  mercury  iodide  studied  by 
Reinders.1 

The  liquid  mixtures  formed  by  melted  mercury  bromide  and 
mercury  iodide  give  on  cooling  a  single  sort  of  mixed  crystals; 
these  are  the  isomorphous  orthorhombic  crystals  of  yellow  mercury 
iodide. 

Let  us  denote  by  1  the  mercury  bromide  and  by  2  the  mercury 
iodide. 

To  the  value  X=Q  corresponds  the  point  F^  (Fig.  87)  whose 
ordinate  OF1  is  the  temperature  of 
fusion  of  mercury  bromide,  that  is  to 
say,  236°;  from  this  point  start  the  two 
curves  C  and  c  which  end  at  the  point 
F2,  of  abscissa  X  =  l,  of  ordinate  OF2 
equal  to  the  temperature  of  fusion  of 
yellow  mercury  iodide,  that  is,  255°; 
the  two  curves  C,  c  meet  at  an  indif- 
ferent point  /  of  minimum  ordinate; 
#  =  216°.l  is  the  ordinate  of  this  point; 
at  this  temperature  the  mixed  crystals 
have  the  same  composition  as  the 
FIG.  87.  liquid  in  whose  presence  they  exist; 

they  contain  0.59  of  a  molecule  of  mercury  bromide  and  0.41  mole- 
cules of  mercury  iodide. 

1  REINDERS,  Zeitschrift  fur  physikalische  Chemie,  v.  32,  p.  494,  1900. 


1X 


MIXED  CRYSTALS.    ISOMORPHOUS  MIXTURES.       275 

221.  Case  in  which  there  may  be  formed  two  kinds  of  mixed 
crystals. — In  a  great  number  of  cases  two  isomorphous  substances 
are  isodimorphous;  they  may  give  birth  to  two  different  kinds  of 
mixed  crystals  which  we  shall  denote  by  the  indices  a  and  ft. 

Concerning  the  transformation  of  mixed  ft  crystals  into  mixed 
a  crystals,  one  may  repeat  almost  textually  what  we  have  said 
regarding  the  transformation  of  mixed  crystals  into  a  liquid  mix- 
ture. 

Take  a  value  of  X  corresponding  to  a  given  composition  and 
suppose  that,  for  this  composition,  the  ft  crystals  are  in  true  equi- 
librium at  low  temperature. 

If  we  raise  gradually  the  temperatures  of  the  mixed  ft  crystals 
whose  composition  is  X,  these  crystals  remain  unaltered  so  long 
as  the  temperature  is  below  T;  the  temperature  exceeding  T,  they 
commence  to  be  transformed  into  a  crystals;  so  long  as  the  tem- 
perature lies  between  T  and  r7  the  system  of  mean  composition  X 
will  be  composed  of  mixed  ft  crystals  and  mixed  a  crystals,  having 
both  a  composition  different  from  X;  finally,  when  the  tempera- 
ture exceeds  T7,  the  system  will  be  entirely  in  the  state  of  a  crystals. 

Let  M  be  the  point  of  abscissa  X  and  of  ordinate  T,  and  //'  the 
point  of  abscissa  X  and  of  ordinate  r7.  When  X  varies,  the  point 
M  describes  a  curve  F,  and  the  point  //  describes  a  curve  f.  The 
points  in  the  plane  located  below  the  curve  F  represent  states 
where  the  system  is  homogeneous  under  the  form  of  ft  crystals ;  the 
plane  situated  above  the  curve  F  represent  states  in  which  the 
system  is  homogeneous  under  the  form  of  a  crystals;  finally,  the 
points  situated  between  the  two  curves  F  and  7-  represent  hetero- 
geneous states  where  the  system  is  formed  of  a  and  ft  crystals. 

The  mixtures  of  mercury  bromide  and  of  mercury  iodide, 
studied  by  Reinders,  furnish  us  with  another  very  simple  example 
of  these  propositions. 

It  is  known  that  when  the  temperature  is  lowered  to  about 
126°,  yellow  mercury  iodide  changes  over  to  the  red  iodide;  simi- 
larly, by  a  lowering  of  temperature,  the  mixed  crystals  of  mercury 
iodide  and  of  mercury  bromide,  which  are  isomorphous  with  the 
yellow  iodide  and  which  play  here  the  role  of  the  a  crystals,  are 
transformed  into  mixed  isomorphous  crystals  of  red  iodide,  play- 
ing the  role  of  the  ft  crystals. 


276  THERMODYNAMICS  AND  CHEMISTRY. 

Reinders  has  drawn  for  these  crystals  the  curves  F  and  f 
which  are  indicated  in  Fig.  87.  These  two  curves  unite  for  X=l 
in  a  point  t2,  whose  ordinate  It2  is  equal  to  the  temperature  of 
transformation  of  yellow  mercury  iodide  into  the  red  iodide,  that 
is,  at  126°.  These  two  curves  do  not  extend  to  the  line  OT  where 
X  =  0;  indeed,  beyond  a  certain  content  in  mercury  bromide,  one 
observes  only  the  mixed  a  crystals. 

In  Fig.  87  the  cross-hatched  regions  correspond  to  the  hetero- 
geneous states  of  the  system;  in  the  region  covered  with  cross- 
hatching  parallel  to  OX  the  system  is  formed  of  liquid  and  mixed 
a  crystals  in  the  region  cross-hatched  parallel  to  OT  the  system 
is  composed  of  a  and  /?  crystals. 

222.  The  two  kinds  of  mixed  crystals  may  be  furnished  by 
the  liquid  mixture.  Case  of  the  transition- point. — In  the  case 
we  have  just  examined  the  transformation  of  the  a  crystals  into 
/?  crystals  is  produced  at  too  low  temperatures  for  the  liquid  mix- 
ture to  be  observable;  the  liquid  cannot  therefore  deposit  other 
than  the  a-salt  crystals,  which  simplifies  the  study  of  these  phe- 
nomena. 

In  a  great  number  of  cases  it  is  quite  otherwise;  the  liquid 
mixture  may,  according  to  circumstances,  furnish  either  the  a^ 
crystals  or  the  a2  crystals;  the  a^  crystals  if  it  contains  a  large 
proportion  of  substance  1,  case  for  which  X  has  there  a  value  near 
to  0;  the  «2  crystals  if  it  contains  a  large  proportion  of  substance 
2,  case  for  which  X  has  a  value  near  to  1. 

Let  us  consider,  for  example,  a  liquid  mixture  obtained  by 
melting  together  silver  nitrate  and  sodium  nitrate,  a  mixture  which 
has  been  studied  by  'Hissink ; *  let  us  give  the  index  1  to  silver 
nitrate,  and  the  index  2  to  sodium  nitrate. 

The  liquid  mixtures  rich  in  silver  nitrate  (X  near  to  0)  furnish 
mixed  at  crystals,  which  are  hexagonal  crystals,  isomorphous  with 
those  which  are  furnished  by  fused  silver  nitrate  at  its  freezing- 
point. 

The  liquid  mixtures  rich  in  sodium  nitrate  (X  near  to  1)  crys- 
tallize in  mixed  a,  crystals,  which  are  also  hexagonal,  but  of  differ- 
ent parameters  from  the  others;  these  crystals  are  isomorphous 

1  HISSINK,  Zeitschrift  fur  physikalische  Chemie,  v.  32,  p.  537,  1900. 


MIXED  CRYSTALS.    ISOMORPHOUS   MIXTURES.       277 

with  those  furnished  by  sodium  nitrate  fused  in  the  pure  state  at 
its  freezing-point. 

To  each  of  these  kinds  of  mixed  crystals  corresponds  a  curve; 
these  two  curves  are  respectively  analogous  to  those  we  have 
called  C  and  c  (Art.  220);  we  shall  call  Ct  and  cx  the  two  curves 
which  refer  to  the  at  crystals;  C2,  c2  the  two  curves  corresponding 
to  the  a2  crystals. 

The  curves  q  and  c2  have  the  appearance  as  shown  in  Fig.  88. 
The  line  cx  rises  from  left  to  right     T 
starting  from  the  point  Flt  whose 
ordinate  OFl  =  2Q8°.Q  is  the  freez- 
ing-point  of   pure  silver   nitrate. 
The  line  c2  descends  from  right  to 
left  beginning  at  F2,  whose  ordi- 
nate  1^2=308°  is   the    freezing- 
point  of  pure  sodium  nitrate. 

These  two  curves  meet  in  a 
point  3  of  ordinate  00 =217°. 5. 

When,  therefore,  the  freezing- 
point  increases  from  0^1  =  208°.6 
to  00  =  217°.5,  the  liquid  mix- 
ture deposits  mixed  crystals  of  the 
at  kind;  when,  exceeding  06= 
217°.5,  the  temperature  of  freezing 


Liquid 


O  2B,xifi    £2     XiBal        X 

FIG.  88. 

increases  to  1F2  =  308°,  the  liquid  furnishes  mixed  crystals  of  the 
a2  kind.  One  may  say  that  the  temperature  0  is  a  transition 
temperature  and  that  the  point  3  common  to  the  two  curves  c1;  c2 
is  a  transition-point. 

The  curve  Cv  starting  from  the  point  Flt  rises  from  left  to 
right  up  to  the  point  Alt  of  ordinate. 00,  remaining  below,  and 
hence  to  the  right  of,  the  line  q;  the  curve  C2,  from  the  point  F2, 
descends  from  right  to  left  as  far  as  the  point  A2,  of  ordinate  00, 
remaining  below,  and  so  to  the  right  of,  the  line  c2;  finally,  the 
point  A2  is  to  the  right  of  the  point  Ar 

If  we  designate  by  H,  £lf  £2,  the  abscissae  of  the  points  3,  A19 
A2,  we  have 


278  THERMODYNAMICS  AND  CHEMISTRY. 

At  the  temperature  6,  ordinate  of  the  point  3,  the  same- 
liquid,  of  composition  8,  may  be  in  equilibrium  either  in  contact 
with  the  mixed  ax  crystals,  of  composition  £1;  or  in  contact  with 
the  mixed  «2  crystals,  of  composition  £2;  the  principles  of  thermo- 
dynamics show  then  that  the  temperature  0,  a1  crystals,  of  com- 
position <?!,  and  a2  crystals,  of  composition  £2,  put  in  presence  of 
each  other,  remain  in  equilibrium;  whence  results  an  important 
property  of  the  points  A1  and  A2. 

Take  a  temperature  T,  less  than  0.  At  this  temperature  one 
may  observe  mixed  at  crystals  which  remain  in  equilibrium  with 
mixed  a2  crystals;  it  is  sufficient  for  this  that  the  a±  crystals  have 
a  composition  X=X1}  and  that  the  a2  crystals  have  a  composition 
X=%2,  %2  being  greater  than  £r 

Let  M!  be  the  point  of  coordinates  (^T),  M2  the  point  of 
coordinates  (%2T);  when  the  temperature  T  is  made  to  change 
keeping  it  less  than  6,  the  point  Ml  describes  a  line  B^M^  and  the 
point  M2  a  line  B2M2. 

From  the  properties  which  we  have  recognized  the  points  A^  and 
A2  to  possess,  the  line  B^M^  passes  through  the  point  A1}  and  the 
line  B2M2  through  the  point  A2. 

It  is  now  easy  to  find  the  properties  possessed  by  the  system 
when  the  position  of  its  representative  point  (XT)  is  known. 

If  the  representative  point  is  above  the  lines  c1;c2,  the  system 
is  in  the  state  of  homogeneous  liquid. 

If  the  representative  point  is  in  the  region  OF^A^  of  the  plane, 
the  system  is  in  the  state  of  mixed  homogeneous  crystals  of  the  a^ 
kind. 

If  the  representative  point  is  in  the  region  F2A2B21}  the  system 
consists  of  homogeneous  crystals  of  the  a2  kind. 

If  the  representative  point  is  located  in  none  of  these  three 
regions,  the  system  of  mean  composition  X  is  heterogeneous. 

It  is  formed  of  liquid  and  of  ax  crystals  if  the  representative 
point  is  in  the  triangle  3FXA  ;  of  liquid  and  a2  crystals  if  the 
representative  point  is  in  the  region  B^A^B.A^. 

223.  Case  of  a  eutectic  point. — The  arrangement  which  we 
have  just  studied  is  not  the  only  one  that  may  be  met  with; 
the  mixtures  of  sodium  and  potassium  nitrates,  also  studied  by 
Hissink,  show  another. 


MIXED  CRYSTALS.    ISOMORPHOUS  MIXTURES.        279 

Let  us  give  the  index  1  to  sodium  nitrate,  and  the  index  2  to 
potassium  nitrate. 

The  mixtures  rich  in  sodium  nitrate  furnish  mixed  «1  crystals, 
which  are  hexagonal,  isomorphous  with  the  crystals  denoted 
by  a2  of  the  preceding  article.  The  mixtures  rich  in  potassium 
nitrate  furnish  mixed  a2  crystals,  which  are  orthorhombic. 

Starting  from  the  point  Flf  whose  ordinate  0^  =  308°  is  the 
freezing-point  of  pure  sodium  ni- 
trate, the  curve  cx  descends  con- 
stantly from  left  to  right  (Fig.  89) ; 
from  the  point  F2,  whose  ordinate 
LF2=337°  is  the  freezing-point  of 
pure  potassium  nitrate,  the  line  % 
descends  constantly  from  right  to 
left. 

These  two  curves  meet  in  a  point 
E,  whose  ordinate  0(9  =  218°  is  less 
than  the  fusing  temperatures  of 
pure  sodium  and  potassium  nitrates  ; 
E  is  the  abscissa  of  the  point  E.  °  BI  ^  ^ 

The  two  curves  Clf  C2,  starting  FlG-  89- 

respectively  from  the  points  Flt  F2,  descend  to  the  points  Alt 
A2,  which  have  the  common  ordinate  06 ;  £t  is  the  abscissa  of 
the  point  Alf  £2  is  the  abscissa  of  the  point  A2}  and  it  follows 
directly  that 


Here  are  the  remarkable  properties  which  such  an  arrangement 
necessitate : 

Let  us  take,  at  a  sufficiently  high  temperature,  a  liquid  mixture 
of  the  substances  1  and  2  and  suppose,  in  order  to  be  definite,  that 
the  composition  of  this  mixture  corresponds  to  a  value  of  X  greater 
than  8.  The  representative  point  is  at  P0.  Lower  gradually 
the  temperature  of  the  system. 

As  long  as  this  temperature  remains  above  a  certain  limit  the 
mixture  will  stay  liquid  and  this  liquid  will  have  an  invariable  com- 
position; the  representative  point  of  the  state  of  the  liquid  will 
follow  the  line  P0P2  parallel  to  TO. 


280  THERMODYNAMICS  AND  CHEMISTRY. 

It  will  thus  attain  the  point  P2,  located  on  the  line  c2;  at  this 
instant  mixed  crystals  of  the  «2  form  begin  to  deposit;  in  order 
to  obtain  the  representative  point  p2  of  the  state  of  these  crystals, 
it  will  suffice  to  draw  a  parallel  to  OX  through  the  point  P2,  until 
it  meets  the  curve  C2.  These  crystals  being  richer  in  potassium 
nitrate  than  the  liquid  from  which  they  come,  their  precipitation 
causes  the  value  of  X  to  decrease  for  the  liquid ;  the  representative 
point  for  the  liquid  state  is  displaced  towards  the  left ;  if  the  cooling 
is  slow  enough  for  the  equilibrium  to  be  at  every  instant  established 
between  the  liquid  and  the  mixed  crystals,  the  representative 
point  for  the  liquid  descends  the  line  cl  and  reaches  the  point  E. 

Consider,  at  the  point  E,  the  liquid  of  temperature  and  com- 
position B. 

As  soon  as  we  decrease  below  6  the  temperature  of  the  system 
whose  mean  composition  is  B,  this  system  must  form  a  hetero- 
geneous mixture  consisting  of  the  mixed  ax  crystals  of  composition 
£t,  and  of  the  mixed  «2  crystals  of  composition  £2;  therefore,  if 
we  continue  to  cool  our  liquid,  it  will  entirely  freeze  furnishing 
such  a  solid  mixture;  this  mixture  is  produced  in  the  same  way 
as  the  eutectic  mixtures  studied  in  Art.  207;  like  them,  it  has  a 
definite  mean  composition ;  like  them,  it  is  a  heterogeneous  mixture 
of  two  kinds  of  crystals;  only  these  crystals,  instead  of  being  of 
definite  chemical  kinds,  are  mixed  crystals;  each  of  the  two  kinds 
of  mixed  crystals  enclosed  in  the  eutectic  mixture  has,  further- 
more, a  fixed  composition. 

We  should  have  reached  analogous  conclusions  by  taking  to 
start  with  a  liquid  whose  composition  would  have  corresponded 
to  a  value  of  X  less  than  B. 

We  shall  say  for  the  case  in  hand  that  the  point  E  is  a  eutectic 
point. 

For  the  case  studied  by  Hissink  the  eutectic  mixture  obtained 
at  218°  had  sensibly  the  chemical  formula 

0.507KNO3  +  0.493NaN03. 

It  was  formed  of  a  conglomerate  of  ax  crystals,  having  the 
formula 

0.24KN03+0.76NaN03, 


MIXED  CRYSTALS.    ISOMORPHOUS  MIXTURES.        281 
and  of  «2  crystals,  having  the  formula 

0.85KNO3+0.15NaNO3. 

224.  Isotrimorphous  and  isotetramorphous  substances; 
studies  of  Hissink  and  of  van  Eyk. — The  silver  and  sodium 
nitrates  studied  by  Hissink  are  isotrimorphous  bodies;  besides 
the  «!  and  «2  erystals  of  Art.  222  which  may  coexist  with  the 
liquid,  one  may  observe  other  mixed  crystals  ft  at  temperatures  at 
which  the  liquid  cannot  exist. 

Pure  silver  nitrate,  hexagonal  at  temperatures  above  159°.5, 
is  orthorhombic  at  temperatures  below  159°.5;  the  mixed  ax 
crystals  are  isomorphous  with  the  hexagonal  si'ver  nitrate;  the 
mixed  ft  crystals  are  isomorphous  with  the  orthorhombic  silver 
nitrate. 

These  mixed  ft  crystals  are  generated,  by  a  sufficient  lowering 
of  temperature,  at  the  expense  of  the  mixed  a^  crystals  very  rich 
in  silver  nitrate. 

One  may,  for  the  transformation  of  at  crystals  into  ft  crystals, 
construct  the  curves  /\,  7-  (Fig.  90),  analogues  of  the  curves  F,  f 
which  were  discussed  in  Art.  221. 

These  curves  start  from  the  point  rlt  whose  abscissa  is  X=0 
and  whose  ordinate  Or1=159°.5  is  the  transformation  tempera- 
ture of  crystals  of  pure  silver  nitrate.  They  both  descend  from 
left  to  right. 

The  line  ft  meets  the  line  A&  in  a  point  Blf  of  ordinate  06l= 
138°;  to  this  ordinate  corresponds  a  point  Dl  on  the  line  7\  The 
point  Dl  is  a  eutectic  point;  by  lowering  the  temperature,  the 
«!  crystals  are  transformed  into  a  mixture  of  ft  and  «2  crystals. 

At  the  temperature  138°  the  at  crystals,  whose  composition 
is  X=61Bl,  remain  in  equilibrium  in  contact  with  theft  crystals, 
whose  composition  is  X=dlBl;  they  remain  also  in  equilibrium 
in  contact  with  the  a2  crystals,  whose  composition  is  X=01B2; 
thermodynamics  shows  without  difficulty  that  the  ft  and  «2  crys- 
tals, whose  composition  we  have  just  stated,  remain  in  equilibrium 
at  138°  in  contact  with  each  other. 

At  temperatures  less  than  138°  one  may  observe  equilibrium 


282 


THERMODYNAMICS  AND  CHEMISTRY. 


states  betwen  the  ft  and  a2  crystals.  The  two  points  which  repre- 
sent the  mixed  ft  and  «2  crystals  capable  of  resting  in  contact 
at  a  given  temperature  have  two  curves  for  loci.  From  what  we 


O  Gj 


G2        IX 


FIG.  90. 


have  just  said,  the  first  of  these  two  curves,  GJ)lt  passes  through 
the  point  Dlt  and  the  second,  G2B2,  ends  at  the  point  B2. 

There  is  thus  obtained  the  arrangement  of  curves  shown  in 
Fig.  90;  in  this  figure  the  scale  is  not  preserved. 

The  admirable  researches  of  van  Eyk  1  on  the  mixtures  of 
potassium  nitrate  and  thallium  nitrate  have  unravelled  a  still  more 
complicated  case,  for  the  salts  considered  are  isotetramorphous; 
Fig.  91,  where  thallium  nitrate  has  been  taken  for  substance  1 


1  VAN  EYK,  Zeitschrift  fur  physikalische  Chemie,  v.  30,  p.  430,  1899;  Ar- 
chives neerlandaises  des  Sciences  exactes  et  naturelles,  2d  S.,  v.  4,  p.  118,  1901. 


MIXED  CRYSTALS.    ISOMORPHOUS  MIXTURES 


283 


and  potassium  nitrate  for  substance  2,  and  where  the  scale  is  not 
kept,  summarizes  the  results  of  these  researches. 

The  liquid  may  coexist  with  the  at  crystals  if  it  is  rich  in  thal- 
lium nitrate  and  with  the  «2  crystals  if  it  is  rich  in  potassium 
nitrate;  the  at  and  «2  crystals  both  belong  to  the  hexagonal  sys- 
tem, but  are  not  isomorphous  with  each  other. 

To  the  freezing  into  «1  crystals  correspond  the  curves  q,  C19 
which  start  from  the  point  Flt  where  1^  =  206°,  and  descend  from 


FIG.  91. 


left  to  right;  to  the  freezing  into  «2  crystals  correspond  the  curves 
c^,  C2,  which  start  from  F2,  of  ordinate  339°,  and  descend  from 
right  to  left. 

The  lines  meet  at  the  point  E,  a  eutectic  point  corresponding 
to  the  temperature  00=182°. 


284  THERMODYNAMICS  AND  CHEMISTRY. 

At  a  lower  temperature  the  a^  crystals  are  changed  into  /?j 
crystals,  which  are  orthorhombic ;  corresponding  to  this  trans- 
formation are  the  curves  ylf  Flf  whicji  start  from  the  point  TI;  of 
ordinate  144°.3,  and  descend  from  left  to  right  to  the  points  Blf 
Dl  of  common  ordinate  00=133°. 

The  temperature  being  lowered  still  more,  the  «2  crystals  are 
changed  into  /?2  crystals,  which  are  orthorhombic.  To  this  trans- 
formation correspond  the  curves  f^T^  starting  from  the  point  T2, 
whose  ordinate  is  129°.5,  these  curves  descend  from  right  to  left 
as  far  as  the  points  D2,  G2,  of  common  ordinate  108  .5. 

These  very  complex  cases,  reduced  to  such  clear  and  expressive 
representati  ns,  are  quite  fitting  to  give  emphasis  to  the  importance 
of  thermodynamical  princip  es  in  the  s  udy  of  isomorphism. 

224a.  Sulphur  and  phosphorus.  Researches  of  Boulouch. — 
The  methods  we  have  just  exposed  help  us  to  decide  if  the  crystals 
which  grow  in  the  presence  of  a  liquid  mixture  are  a  definite  com- 
pound or  mixed  crystals;  they  are  also  valuable  for  the  discussion 
of  certain  questions  in  litigation;  in  particular  they  seem  called 
to  play  a  considerable  role  in  the  study  of  systems  where  two 
metalloids  exist  togeth  r;  more  than  one  substance,  obtained  in 
such  conditions  and  regarded  as  a  definite  ompound,  is  perhaps 
only  a  conglomeration  of  mixed  crystals. 

Boulouch  l  has  applied  this  method  to  the  study  of  bodies 
which  are  formed  within  a  liquid  mixture  of  sulphur  and  phos- 
phorus. 

Berzelius  had  described  the  sulphides  P4S,  P2S,  P2S12  as  being 
formed  in  such  conditions ;  according  to  the  researches  of  Boulouch, 
none  of  these  definite  compounds  really  exists.  By  cooling  liquid 
mixtures  of  sulphur  and  phosphorus,  only  two  kinds  of  mixed 
crystals  are  obtained. 

The  first  are  fovmed  in  the  mixtures  rich  in  phosphorus;  they 
are  isomorphous  with  the  crystals  of  white  phosphorus;  the  freez- 
ing-point of  the  liquid  giving  rise  to  these  crystals  s  the  lower  as 
the  liquid  is  richer  in  sulphur;  the  fusing-point  of  these  crystals 
is  also  the  lower  as  the  crystals  contain  more  sulphur. 

If  the  temperature  is  taken  as  ordinate,  and  for  abscissa  the 

1  R.  BOULOUCH,  Comptes  Rendus,  v.  135,  p.  166,  1902. 


MIXED  CRYSTALS.    ISOMORPHOUS  MIXTURES.        285 


ratio  X  of  the  mass  of  sulphur  to  the  total  mass  of  the  mixture, 

we  obtain  (Fig.  B)  a  congelation  curve  T 

which  descends  from  left  to  right  along 

PE  and  a  fusion  curve  which  descends 

along  PA ;  P  is  the  freezing-point  of  pure 

phosphorus. 

The  second  mixed  crystals  are  formed 
hi  the  mixtures  rich  in  sulphur;  they  are 
isomorphous  with  the  crystals  of  clino- 
rhombic  sulphur;  the  congelation  curve 
descends  from  right  to  left  along  SE,  and 
the  fusion  curve  along  SB;  S  is  the  freez- 
ing point  of  clinorhombic  sulphur.  o  <r  i  x 

The  point  E  furnishes  us  with  an  ex-  FIG  B. 

ample  of  eutectic  point  as  sha  p  as  that  studied  by  Hissink  (Art. . 
223);  the  co-ordinates  o!  this  point  are  o= 0.228  and  0  =  9°.8. 

The  temperature  9°.8  is  the  fusing-point  of  every  solid  system 
containing  the  two  kinds  of  mixed  crystals  at  once. 

The  liquid  mixture  remains  readily  in  surfusion  with  respect 
to  the  mixed  crystals  of  the  second  k  nd ;  it  may  then  furnish 
mixed  crystals  of  the  first  kind;  the  corresponding  freezing-points 
are  located  on  the  line  PY,  extension  of  the  line  PE. 

224b.  Sulphur  and  selenium.  W.  E.  Ringer's  researches. — 
The  conclusions  reached  by  Boulouch  studying  the  mixtures  of 
sulphur  and  phosphorus  are  remarkably  simple;  much  more  com- 
plicated results  follow  the  study  o"  the  mixtures  of  sulphur  and 
selenium,  as  has  been  found  recently  by  W.  E.  Ringer.1 

There  are  here  formed  four  k  nds  of  mixed  crystals ;  the  various 
peculiarities  of  their  fusion,  their  crystallization,  their  trans- 
formation into  each  other  are  represented  in  Fig.  C,  where  the 
scale  has  not  been  exactly  kept  Temperatures  are  taken  as  or- 
dinates ;  as  abscissae  are  taken  the  ratio  X  of  the  mass  of  selenium 
to  the  total  mass  of  the  mixture;  the  full  lines  have  been  deter- 
mined experimentally. 

Two  kinds  of  crystals  rich  in  sulphur  may  be  obtained;  we 
shall  denote  them  by  the  letters  a  and  /?;  the  a  crystals  are  iso- 


W.  E  RINGER,  Zeitschrift  fur  anorganiscke  Chemiv,  v.  32,  p.  183,  1902. 


286 


THERMODYNAMICS  AND  CHEMISTRY. 


morphous  with  orthorhombic  sulphur,  and  the  /?  crystals  with 
clinorhombic  sulphur. 

The  line  AE  is  the  congelation  line  of  the  liquid  into  the  state 
of  @  crsytals;  the  line  AB  is  the  fusion  line  of  these  crystals;  the 
point  A  is  the  fusing-point  of  clinorhombic  sulphur. 


FIG.  C. 

The  mixed  /?  crystals  may,  when  they  are  cooled,  be  trans- 
formed into  mixed  a  crystals  the  trans  formation -points  form 
the  line  CF;  inversely,  when  they  are  heated,  the  mixed  a  crystals 
are  transformed  into  /?  crystals,  the  transformation-points  lie  on 
the  line  CD;  C  is  the  transformation-point  of  cl  norhombic  into 
orthorhombic  sulphur. 

The  liquid  mixtures  very  rich  in  selenium  furnish  mixed  d 
crystals.,  isomorphous  with  metallic  selenium;  the  corresponding 
freezing-points  form  the  line  GI]  the  fusing-points  of  the  d  crystals 
form  the  line  GH ;  G  is  the  fusing-point  of  metallic  selenium. 

Finally,  the  liquid  mixtures  of  mean  composition  deposit 
mixed  crystals  of  a  fourth  form,  7-,  at  the  congelation-points,  which 


MIXED  CRYSTALS.    ISOMORPHOUS  MIXTURES.        287 

are  those  of  the  line  El;  the  fusing  points  of  these  7-  crystals  are 
those  of  the  line  MK. 

According  to  the  position,  in  the  plane,  of  the  point  representing 
the  temperature  and  composition  of  the  system,  the  system  in 
equilibrium  may  contain  a  single  one  of  the  five  phases  a,  /?,  7-,  d, 
and  L  (liquid),  or  be  divided  into  two  of  these  phases.  The  various 
circumstances  which  may  occur  have  been  marked  on  Fig.  C. 

The  researches  of  Boulouch  and  of  Ringer  indicate  clearly  that 
the  whole  chemistry  of  the  metalloids  should  be  submitted  to 
a  revision  guided  by  the  methods  of  Thermodynamics. 


CHAPTER  XIV. 

MIXED   CRYSTALS    (Continued).     OPTICAL    ANTIPODES.     METAL- 
LIC ALLOYS. 

I.  OPTICAL  ANTIPODES. 

225.  Mixed  crystals  are  not  limited  to  mixtures  of  isomor- 
phous  bodies.  Their  frequency  in  organic  chemistry. — Mixed 
crystals  are  constantly  met  with  when  there  are  crystallized  to- 
gether two  substances  of  similar  chemical  formulae,  isomorphous 
in  the  sense  Mitscherlich  gave  to  this  term.  But  frequently  also 
substances  which  have  not  similar  chemical  formulae  show  them- 
selves capable  of  forming  mixed  crystals.  Thus  we  have  seen,  in 
Art.  219,  ferric  chloride  forms  mixed  crystals  with  ammonium 
chloride.  Facts  of  this  sort  indicate  that  prudence  is  needed 
when  use  is  made  of  Mitscherlich's  law  in  the  appreciation  of 
chemical  analogies;  the  property  of  giving  mixed  crystals  often 
accompanies  the  similarity  of  chemical  formulae,  but  it  may  be 
met  with  when  this  similarity  is  in  default. 

The  compounds  of  organic  chemistry,  and  especially  the  sub- 
stances in  the  aromatic  series,  are,  in  a  great  number  of  cases, 
capable  of  forming  mixed  crystals  two  by  two.  This  property  is 
often  correlative  of  a  true  crystallographic  isomorphism;  this  is 
what  takes  place,  for  example,  with  azobenzol  and  stilbene, 
studied  from  this  point  of  view  by  G.  Bruni.1  Furthermore,  the 
symbols  of  these  two  substances 

N— C0H5  HC— C6H5 

N-C6H5  HC-C6H5 

Azobenzol  Stilbene 


1  G.  BRUNI,  Rendiconli  dell'  Accademia  del  Lincei,  v.  8,  p.  570,  1899. 

288 


I       >NH 


MIXED  CRYSTALS.    OPTICAL  ANTIPODES.  289 

may  be  regarded  as  analogues,  so  that  here  is  a  case  of  complete 
isomorphism,  in  the  sense  given  to  this  term  by  Mitscherlich. 

In  other  cases  it  is  more  difficult  to  admit  of  an  analogy  be- 
tween the  chemical  symbols  of  substances  which  mix  in  crystal- 
lizing; it  is  thus1  that  carbazol  and  anthracene  both  form  mixed 
crystals  with  phenanthrene,  while  the  chemical  symbols  of  these 
three  substances, 

CH 

HA-CH  /  \ 

II  H4C6     H 

HA-CH 

CH 

Phenanthrene  Anthracene  Carbazol 

can  with  difficulty  be  regarded  as  analogous. 

The  absence  of  analogy  is  still  more  striking  between  naphtha- 
lene and  monochloracetic  acid,  whose  mixtures  have  been  studied 
by  Cady.2  Within  these  mixtures  there  are  formed  two  kinds  of 
mixed  crystals;  the  first,  rich  in  naphthalene,  are  isomorphous 
with  pure  naphthalene  crystals;  the  others,  rich  in  monochloracetic 
acid,  are  isomorphous  with  those  furnished  by  this  acid  taken  by 
itself. 

The  observed  phenomena  have  the  same  characteristics  as 
those  described  in  Art.  223.  There  .we  found  a  eutectic  con- 
glomerate whose  mean  composition  is  fixed,  and  which  is  com- 
posed of  two  kinds  of  mixed  crystals. 

Organic  chemistry  furnishes  innumerable  examples  of  mixed 
crystals,  among  which  several  have  been  studied  by  Kiister, 
Garelli,  Bruni,  and  various  other  observers.3 

226.  Optical  antipodes.  Inactive  substances  to  which  they 
may  give  rise.—  The  idea  of  mixed  crystals  assumes  great  impor- 
tance in  the  discussions  relative  to  the  properties  of  substances 

1  G.  BRUNI,  Rendiconti  d*W  Accademia  dei  Lined,  v.  7,  p.  138,  1898. 

3  CADY,  Journal  of  Physical  Chemistry,  v.  3,  p.  127,  1899. 

3  The  reader  will  find  interesting  information  on  the  whole  of  this  question 
of  solid  solutions  in  the  following:  G.  BRUNT,  Ueber  feste  Losungen  (Arhen's 
Sammlung,  v.  6,  part  12). 


290  THERMODYNAMICS  AND   CHEMISTRY. 

gifted  with  rotary  power,  discussions  essential  to  the  progress  of 
stereochemical  doctrines. 

Everybody  is  acquainted  with  the  researches  of  Pasteur  on  the 
tartaric  acids  and  the  tartrates. 

There  exist  two  tartaric  acids  which  possess  exactly  the  same 
physical  and  chemical  properties  save  one:  solutions  of  the  first 
possess  a  certain  rotary  power  to  the  right ;  solutions  of  the  second 
possess  exactly  the  same  rotary  power,  but  to  the  left.  The  first 
is  the  right-handed  acid,  the  second  is  the  left-handed  acid. 

The  crystals  furnished  by  the  right-handed  acid  do  not  possess 
the  rotary  power,  but  they  have  a  non-superposable  hemiedry; 
the  crystal  is  not  superposable  upon  its  image  in  a  mirror. 

The  left-handed  acid  likewise  furnishes  crystals  without  action 
on  polarized  light  and  possessing  a  similar  hemiedry.  A  left- 
handed  crystal  is  superposable  upon  the  image  of  a  right-handed  crys- 
tal in  a  mirror  and  reciprocally. 

By  representing  each  atom  of  quadrivalent  carbon  in  the  form 
of  a  regular  tetrahedron,  stereochemical  notation  attributes  to 
these  two  acids  two  symbols  as  distinct  as  the  crystals  giving  them. 
The  symbol  of  the  right-handed  acid  is  not  superposable  upon 
itself  in  a  mirror,  but  on  reflecting  it  in  a  mirror  it  reproduces  the 
formula  of  the  left-handed  acid. 

These  two  acids  have  an  isomer,  the  inactive  acid,  whose  solutions 
are  without  action  on  polarized  light;  the  crystals  which  it  fur- 
nishes are  holoedrons;  each  of  them  is  superposable  upon  its  image 
seen  in  a  mirror;  stereochemistry  attributes  to  this  inactive  acid 
a  formula  which  is  reproduced,  identical  with  itself  by  reflection  in 
a  plane  mirror;  no  reaction  separates  this  acid  into  right-  and  left- 
handed  acids. 

By  combining  molecule  to  molecule,  right-handed  tartaric  acid 
and  left-handed  tartaric  acid  form  a  polymer,  racemic  acid.  This 
acid,  whose  stereochemical  formula  is  then  superposable  upon  its 
image  in  a  mirror,  gives  holoedric  crystals  gifted  with  the  same 
property;  by  dissolving  it,  one  obtains  a  liquid  devoid  of  rotary 
power. 

These  properties  are  not  peculiar  to  the  tartaric  acids  and  the 
tartrates;  a  great  number  of  organic  compounds  likewise  possess 
them. 


MIXED  CRYSTALS.    OPTICAL  ANTIPODES.  291 

Such  a  compound  possesses  two  isomeric  varieties  which  have 
exactly  the  same  chemical  and  physical  properties  except  one:  the 
right-handed  kind,  in  the  state  of  fusion  or  of  solution,  possesses  a. 
right-handed  power  of  rotation;  the  left-handed  sort  has  exactly 
the  same  rotary  power,  but  to  the  left.  The  two  varieties  of  crys- 
tals are  hemiedric;  the  crystals  of  the  right-handed  variety,  on 
reflection  in  a  mirror,  reproduce  crystals  of  the  left-handed  sort 
and  conversely.  These  crystals  are  in  general  devoid  of  rotary 
power;  when  they  are  so  gifted,  the  right  and  left  crystals  have 
equal  rotary  powers,  in  opposite  directions.  The  stereochemical 
notation  attributes  different  formulae  to  these  two  isomers;  one 
of  the  formulas  is  the  image  of  the  other  in  a  mirror.  These  two 
isomeric  substances  are  said  to  be  enantiomorphous,  or  to  be  optical 
antipodes  of  each  other. 

Often  there  is  occasion  to  add  a  third  inactive  isomer  to  these 
two  optical  antipodes;  devoid  of  rotary  power  in  all  its  states, 
this  inactive  isomer  furnishes  holoedral  crystals;  stereochemistry 
assigns  it  a  formula  which  is  reproduced,  identical  with  itself,  by 
reflection  in  a  mirror. 

In  a  great  number  of  cases  a  molecule  of  the  right  isomer  may 
combine  with  a  molecule  of  the  left  isomer  to  form  a  polymer 
which  is  without  action  on  polarized  light  and  which  gives  holoedral 
crystals;  by  analogy  with  racemic  acid  and  the  racemates,  which 
are  formed  in  this  way,  the  name  given  to  this  polymer  is  racemic 
compound. 

The  racemic  combination  is  not  the  only  solid  substance  which, 
on  melting  or  dissolving,  furnishes  a  liquid  inactive  by  compensation. 
The  same  property  belongs  to  a  mixture  of  right  and  left  crystals 
where  the  two  kinds  of  crystals  appear  in  equal  quantities.  It 
belong  likewise  to  mixed  crystals,  which  the  two  left  and  right 
varieties  are  often  capable  of  furnishing,  when  the  two  sorts  figure 
in  the  same  proportion  in  these  mixed  crystals. 

Not  only  may  the  two  optical  antipodes  furnish  mixed  crystals, 
but  it  also  happens  that  each  of  them  may  furnish  mixed  crystals 
with  the  inactive  isomer.  Thus  Fock  *  has  made  the  following 
curious  observation: 

1  FOCK,  Zeitschrift  fur  KrystaUographie,  v.  31,  p.  479,  1899. 


292  THERMODYNAMICS  AND  CHEMISTRY. 

Inactive  pinonic  acid  (pinonsaure),  which  is  orthorhombic, 
forms,  either  with  right  pinonic  acid  or  with  left  pinonic  acid, 
mixed  orthorhombic  crystals.  It  forms  also,  with  right  pinonic 
acid,  mixed  quadratic  crystals  having  hemiedry,  and  rigorously 
isomorphous  with  the  crystals  given  by  right  pinonic  acid  when 
it  is  isolated.  Finally,  it  gives,  with  left  pionoic  acid,  mixed  crys- 
tals symmetrical  with  the  preceding 

A  racemic  compound  may  perhaps  be  formed  of  the  crystals 
mixed  with  each  of  the  two  optical  antipodes,  although  as  yet 
the  fact  has  not  been  demonstrated  with  certainty. 

227.  Freezing  of    the   mixture  of  two  optical  antipodes. — 
Suppose  that  two  substances,  optical  antipodes  of  each  other,  are 
melted  and  mixed  together.     Let  us  study  the  freezing-point  of 
this  mixture  and  the  nature  of  the  precipitate  obtained. 

To  express  the  composition  of  the  liquid  mixture  or,  if  there  is 
need,  of  the  precipitate  obtained,  we  shall  Jay  off  on  the  axis  of 
abscissae  the  mass  X  of  the  left  antipode  contained  in  a  unit  mass 
of  the  mixture;  (1—X)  will  be  the  mass  of  the  right  antipode 
which  is  associated  with  it.  On  the  axis  of  ordinates  lay  off  the 
temperature  T. 

The  two  right  and  left  antipodes  have  exactly  the  same  physical 
properties;  if,  therefore,  the  liquid  mixture  which  contains  X 
grammes  of  the  right  antipode  and  (1—  X)  grammes  of  the  left 
has  a  certain  freezing-point,  the  liquid  mixture  containing  X 
grammes  of  the  left  antipode  and  (l-X)  of  the  right  should 
have  identically  the  same  freezing-point.  The  freezing-point 
curves  will  therefore  be  symmetrical  with  respect  to  the  line 

3>i 

If  the  system  can  furnish  mixed  crystals,  the  freezing-point 
curve  of  these  crystals  will  have  the  same  axis  of  symmetry. 

228.  The  congelation  of  the  mixture  furnishes  neither  racemic 
compound  nor  mixed  crystals. — This  is  the  simplest  case. 

The  mixtures  rich  in  he  right-handed  antipode  should  deposit 
crystals  which  enclose  exclusively  this  right  antipode;  the  phe- 
nomena may  be  compared  in  all  respect  to  the  formation  of 
ice  within  a  salt  solution.  The  freezing-point  is  the  lower  as 
the  richness  of  the  liquid  mixture  in  the  left  antipode  is  the 
greater. 


MIXED  CRYSTALS.    OPTICAL  ANTIPODES. 


293 


The  freezing-point  curve  of  right  crystals  ((Fig.  92)  starts 
from  the  point  Flt  fusing-point  of  the  right 
crystals  in  the  pure  state,  and  descends 
from  left  to  right.  The  congelation  curve 
of  the  left  crystals  starts  from  F2,  fusing- 
point  of  the  pure  left  crystals;  the  two 
points  Fv  F2  have  the  same  ordinate  equal 
to  the  common  temperature  of  fusion  of 
the  right  and  left  crystals.  ^  1 x 

These  two  curves  intersect  at  a  point  E,  FIG.  92. 

of  abscissa  J  and  ordinate  0]  it  is  a  eutectic  point  analogous  to 
that  observed  (Art.  214)  in  studying  the  congelation  of  a  mixture 
of  two  melted  salts  which  do  not  form  a  double  salt;  the  two  cases 
diffe  from  one  another  only  in  he  aspect  of  the  two  fusion  curves 
which  are  anything  for  the  case  tre  ted  in  Art.  214,  and  both  sym- 
metrical with  respect  to  the  line  X=%  for  the  present  case.  The 
eutectic  conglomerate  has  for  mean  composition  X=$;  it  encloses 
in  equal  proportions  right  and  left  ciystals;  melted  or  dissolved, 
it  will  give  an  inactive  mixture  by  compensation. 

This  case,  theoretically  possible,  does  not  seem  to  have  been 
met  with  as  yet  among  those  which  have  been  carefully  studied. 

229.  The  congelation  of  the  mixture  may  give  a  racemic  com- 
pound.— We  encounter  here  a  particular  case  of  the  problem  treated 
in  Art.  214:  freezing  of  a  mixture  of  two  melted  salts  capable  of 
giving  a  double  salt;  the  symmetry  of  the  congelation  curves 
with  respect  to  the  line  X=%  alone  distinguishes  this  from  the 
general  case. 

The  liquid  mixtures  which  contain  a  large  proportion  of  the 
right  antipode  deposit  this  substance 
in  the  pure  state;  on  obtains  a  fusion 
curve  F  El  which  descends  rom  left  to 
right  (Fig.  93);  it  is  a  portion  of  the 
line  FJ2  drawn  in  Fig  92. 

Similarly,   the  liquid  mixtures  rich 
in  left  antipode  deposit  this  substance 
in  the  pure  state;   one  obtains  a  fusion 
curve   F2E2   descending   from   right   to 
left;  it  is  a  portion  of  the  line  F2E  of  Fig.  92. 


294 


THERMODYNAMICS  AND  CHEMISTRY. 


The  two  points  Elt  E2  are  joined  to  another  by  the  congelation 
curve  E1IE2  of  the  racemic  compound;  symmetrical  with  respect 
o  the  line  X=%,  this  curve  has,  for  the  abscissa  X=%,  a  point  / 
where  the  tangent  is  parallel  to  OX;  this  point  /  is  an  indifferent 
point;  the  liquid  mixture  there  has  the  same  composition  as  the 
racemic  compound;  the  ordinate  6  of  this  point  is  the  fusing- 
point  of  this  compound. 

This  disposition  is  frequent;  it  is  met  with  notably  in  the  study 
of  methylbenzoic  ether,  benzylaminosuccinic  acid,  aminosuccinic 
acid, 1  benzoyltetrahydroquinaldine. 2 

In  certain  cases  the  freezing-point  curve  EJE2  of  the  racemic 
compound  is  extremely  reduced  and  the  arrangement  of  Fig.  94 
is  obtained.  There  is  a  tendency  towards  the  case  studied  in  the 


o  ix 

FIG.  94. 


1  X 


FIG.  95. 


preceding  article.  Phenylglycolic  acid  (Mandelsaure)  and  di- 
methylic  ether  of  diacetyltartic  acid,  studied  by  Adriani,  are  two 
examples  of  this. 

In  other  cases  the  two  curves  F&,  F2E2  (Fig.  95)  are  greatly 
reduced  and  the  freezing  curve  E  IE2  of  the  racemic  compound 
occupies  almost  the  whole  field  of  congelation;  this  is  also,  accord- 
ing to  Adriani,  the  case  with  dimethylic  ether  of  tartric  acid. 

230.  The  congelation  of  the  mixture  gives  mixed  crystals. — 
The  freezing  of  the  mixture  may  give  mixed  crystals  whose  every 
element  contains  the  right-handed  substance  and  its  left-handed 
isomer  united  in  a  certain  proportion;  let  x  be  the  mass  of  the 


1  CENTNERSZWER,  Zeitschrift  fur  physikalische  Chemie,  v.  29,  p.  75,  1899. 

2  ADRIANI,  Zeit.  /.  phys.  Chemie,  v.  33,  p.  453,  1900. 


MIXED  CRYSTALS.     OPTICAL  ANTIPODES. 


295 


right  substance  and  (1—  x)  the  mass  of  the  left  substance  in  a  unit 
of  mass  of  these  mixed  crystals;  for  £=J  the  crystals  will  be 
holoedral;  by  fusion  or  solution  they  will  give  a  substance  inactive 
by  compensation;  for  two  values  of  x  equidistant  from  ^  there 
will  be  two  non-superposable  crystalline  forms  both  symmetrical 
with  respect  to  a  plane. 

Let  T  be  the  freezing-point  of  the  liquid  of  composition  X,  and 
M  the  point  of  coordinates  X,  T  (Fig.  96) ;  at  this  temperature  T 
the  liquid  of  composition  X  deposits  mixed  crystals  of  composi- 
tion x\  let  m  be  the  point  of  coordinates  x,  T7. 


O  X  x 


FIG.  96. 


1X0  i  IX. 

FIG.  97. 


While  X  varies  from  0  to  1,  the  point  M  describes  the  curve  C, 
and  the  point  m  the  curve  c,  drawn  below  the  curve  C.  These 
two  curves  pass  through  the  fusing-point  Ft  of  the  right  crystals 
taken  in  the  pure  state  and  through  the  fusing-point  F2  of  the  left 
crystals  in  the  pure  state. 

The  curve  C,  formed  necessarily  by  two  branches  both  sym- 
metrical with  respect  to  the  line  X=%,  has,  for  the  abscissa  X= J, 
a  point  of  maximum  ordinate.  From  the  first  theorem  of  Gibbs 
and  Konovalow  (Art.  iQ4\  which  may  be  applied  to  the  double 
mixture  formed  by  the  mixed  crystals  and  the  mixed  liquid,  this 
point  belongs  also  o  the  line  c,  for  which  it  is  also  a  point  of  maxi- 
mum or  minimum  ordinate.  At  this  indifferent  point  I  the  mixed 
liquid,  which  is  inactive  by  compensation,  must  give,  on  freezing, 
mixed  holoedral  crystals  of  composition  x  =  \. 

According  to  Adriani,  this  disposition  is  observed  in  the  freezing 
of  carvoxime,  bihydrocarvoxime,  and  benzoic  oxime. 


296  THERMODYNAMICS  AND   CHEMISTRY. 

Camphoric  oxime  offers  a  very  curious  particular  case,  repre- 
sented in  Fig.  97.  Whatever  is  the  composition  X  of  the  liquid 
mixture,  Adriani  found  its  freezing-point  constant  and  equal  to 
118°.8;  the  line  C  is  here  reduced  to  a  straight  line  FJ?2  parallel 
to  OX. 

Each  of  the  points  on  this  line  may  be  regarded,  if  so  wished, 
as  a  point  of  maximum  ordinate;  the  theorem  of  Gibbs  and 
Konovalow  may  be  applied  to  each  of  these  points;  whatever  the 
composition  of  the  mixed  liquid,  it  deposits  mixed  crystals  of  the 
same  composition. 

We  have  here  an  example  of  the  rule  that  various  authors 
thought  general  for  the  congelation  of  mixed  crystals  (Art.  220). 

Another  peculiarity  renders  this  example,  studied  by  Adrianir 
very  interesting:  when  the  temperature  is  lowered  the  mixed 
crystals  are  seen  to  transform  themselves  into  crystals  of  a  racemic 
compound;  we  may  construct  a  portion  RJR2  of  the  curve,  analo- 
gous to  the  congelation  curve  of  a  racemic  compound  within  a 
mixed  liquid,  which  corresponds  to  this  transformation;  the  high- 
est point  /  of  his  curve  corresponds  to  the  temperature  103°. 

231.  Formation  in  solution  of  a  racemic  compound. — The 
precipitation  within  a  solution  of  one  of  the  substances  we  have 
just  studied  leads  to  the  study  of  the  equilibrium  of  a  system  no 
longer  bi variant,  but  tri variant;  this  study  is,  from  the  experi- 
mental point  of  view,  much  less  advanced  than  the  preceding;  it 
has  given  rise  nevertheless  to  several  interesting  researches ;  among 
this  number  is  the  analysis  of  the  conditions  of  formation  of  the 
double  racemate  of  sodium  and  ammonium,  analysis  for  which 
we  are  indebted  to  Van't  Hoff  and  van  Deventer.1 

The  formation,  within  a  solution,  of  a  racemate  at  the  expense 
of  the  right  and  left  tartrates  is  comparable  in  all  points  with  the 
formation  of  a  double  salt  at  the  expense  of  two  simple  salts,  forma- 
tion which  we  have  already  studied  (Arts.  102  et  seq.).  The  study 
of  the  phenomenon  will  be  somewhat  simplified  on  account  of  the 
identity  which  exists  between  the  physical  properties  of  the  two 
right  and  left  isomers. 

1  VAN'T  HOFF  and  VAN  DEVENTER,  Zeitschrift  fur  physikalische  Chemie, 
v.  16,  p.  173;  VAN'T  HOFF,  GOLDSCHMIDT,  and  JORISSEN,  ibid.,  v.  17,  p.  49. 


MIXED  CRYSTALS.    OPTICAL  ANTIPODES. 


297 


Take,  as  was  done  in  studying  the  double  salts,  three  axes  of 
rectangular  coordinates  OT,  Os1}  Os2  (Fig.  98) ;  on  the  first  lay  off 
temperatures,  on  the  second  concentrations  of  the  right-handed 
tartrate  solution,  on  the  third  the  concentrations  of  the  left  tar- 
trate  solution. 

We  shall  be  led  to  represent  all  the  possible  equilibrium  states 
by  a  surface  formed  by  the  domain  D  of  the  right  tartrate,  the 
domain  G  of  the  left  tartrate,  the  domain  R  of  the  racemate;  this 
figure  will  be  symmetrical  with  respect  to  the  bisecting  plane  of 
the  diedral 


FIG.  98.  FIG.  99. 

The  double  racemate  of  sodium  and  ammonium  is  formed  at 
the  expense  of  the  two  double  tartrates,  according  to  the  formula 

NaNH4C4H4O6-4H2O  (D)  +  NaNH4C4H4O6  •  4H2O  (G) 

=  Na,(NH4)2(C4H406)  •  2H20  +  4H20. 

This  racemate  forms  in  the  solution  only  at  temperatures 
higher  than  24°;  at  temperatures  below  24°  the  surface  pos- 
sesses only  the  domains  of  the  right  tartrate  and  of  the  left  tar- 
trate. 

The  two,  right  and  left,  tartrates  of  rubidium  furnish  a  race- 
mate  according  to  the  formula  1 


Rb2C406H4 


,  (GO+4H20=Rb4(C4OeH4)2.4H20. 


IVAN'T  HOFF  and  MULLER,  Berichte   der  Deutschen  Chemischen  Gesett- 
schaft,  v.  31,  p.  2206. 


298  THERMODYNAMICS  AND  CHEMISTRY. 

Within  the  solution  this  racemate  is  formed  only  at  tempera- 
tures below  40°.4;  at  higher  temperatures  the  surface  possesses 
only  the  domains  of  the  right  and  left  tartrates  (Fig.  99). 

II.  THE  METALLIC  ALLOYS. 

232.  Liquid  mixtures  which  deposit  metals  in  the  pure  state 
or  a  definite  compound. — The  principles  developed  in  the  pre- 
ceding chapters,  and  particularly  the  notion  of  mixed  crystals, 
commence  to  throw  some  light  on  the  constitution,  obscure  for  so 
long  a  time,  of  the  metallic  alloys;  the  majority  of  alloys  which 
were  regarded  as  definite  chemical  compounds,  having  a  fixed 
composition  and  a  definite  fusing-point,  are  considered  to-day  as 
eutectic  conglomerates  formed  either  of  two  solids,  crystallized  or 
not,  or  of  two  kinds  of  mixed  crystals,  or  of  two  solid  solutions. 

G.  Charpy  l  has  studied  with  the  greatest  care  (Art.  106)  the 
alloy  formed  by  lead,  tin,  and  bismuth;  he  has  also  studied,  but 
in  less  detail,  the  following  ternary  alloys: 

Sn,Cu,Sb; 
Sn,Pb,Sb; 
Pb,Cu,Sb; 

Zn,Sn,Sb; 
Cu,Sn,Pb. 

Besides  these  cases,  the  only  alloys  which  have  been  minutely 
studied  are  mixtures  of  two  metals  which  we  shall  indicate  by 
the  indices  1  and  2. 

The  most  simple  case  to  be  met  with  is  that  where  a  lowering 
of  temperature  imposed  on  the  melted  mixture  of  two  metals 
always  produces  either  the  deposit  of  the  metal  1  in  the  pure  state, 
or  the  deposit  of  the  metal  2  in  the  pure  state. 

This  case  is  quite  similar  to  the  one  treated  in  Art.  214,  where 
a  liquid,  formed  by  two  fused  salts,  can  furnish  no  other  solid 
than  one  or  the  other  salt  in  the  pure  state. 

1  G.  CHARPY,  Etudes  sur  les  alliages  blancs  dits  antifriction  (Contribution 
d  I'Etude  des  alliages,  published  by  the  alloys  committee  of  the  Societe 
d' Encouragement  pour  1'industrie  nationale,  p.  203,  Paris,  1901).  This  is 
a  very  valuable  contribution  to  our  knowledge  of  alloys. 


MIXED  CRYSTALS.     THE  METALLIC  ALLOYS. 


299 


As  abscissa  (Fig.  100)  take  the  value  cf  X  which  represents 
the  composition  of  the  liquid  mixture; 
as  ordinate,  take  the  temperature. 

The  congelation  curve  Cl  of  the  metal  F, 
1  starts  from  the  point  Flf  whose  ordi- 
nate is  the  f  using-point  of  this  metal,  and 
descends  from  left  to  right;  the  congela- 
tion curve  C2  of  the  metal  2  starts  from 
the  point  F2,  whose  ordinate  is  the  fusing- 


o 


I         i 
FIG.  100. 


point  of  this  metal,  and  descends  from 
right  to  left.  These  two  curves  inter- 
sect in  a  eutectic  point  E,  of  abscissa  £  and  ordinate  6;  £  and 
6  indicate  the  composition  of  the  eutectic  conglomerate  and  its 
fusing-point 

Guthrie  *  has  studied  several  systems  which  enter  into  this 
category;  here  are  the  values  he  has  found  for  the  coordinates 
f  and  0  of  the  eutectic  points: 


Mixed  Metals. 

9 

? 

1    Bismuth 

) 

2    Zinc 

248° 

0.0715 

1    Bismuth 

) 

2    Tin 

f 

133° 

0.539 

1    Bismuth 

) 

2    Lead 

122°.  7 

0.4442 

1.  Bismuth.  .  .  . 

) 

2.  Cadmium  

[ 

144° 

0  .  4081 

Also  belonging  to  this  class  is  the  system,  studied  by  Sir 
Roberts- Austen,2  formed  of  the  two  metals  lead  (1)  and  tin  (2); 
for  this  case 

0=183°,  £=0.62. 

But  among  the  systems  of  this  kind,  none  doubtless  has 
been  studied  with  as  much  care  as  the  alloy  formed  of  the  two 


1  GUTHRIE,  Philosophical  Magazine,  5th  S.,  v.  22,  p.  46,  1884. 

*  ROBERTS- AUSTEN,  Proceedings  of  the  Royal  Society,  v.  63,  p.  452,  1898. 


300  THERMODYNAMICS  AND  CHEMISTRY. 

metals  lead  (1)   and  antimony  (2),   object  of  the  researches  of 
Roland-Gosselin,  H.  Gautier,1  and  of  Charpy.2 

When  the  composition  of  the  liquid  mixture  varies  from  X=Q 
to  £  =  0.13,  the  freezing-point  is  lowered  from  T0=326°,  the  freez- 
ing-point of  pure  lead,  to  0  =  228°. 

When  the  composition  of  the  liquid  mixture  changes  from 
£=0.13  to  X=  1,  the  freezing-point  'rises  from  0=228°  to  T1  =  632°> 
freezing-point  of  pure  cadmium. 

The  point 

0=228°,  £=0.13 

is  a  eutectic  point. 

When  a  liquid  mixture  for  which  X  is  included  between  0  and 
0.13  is  brought  to  the  freezing-point,  it  furnishes  crystals  of  pure 
lead;  the  proportions  X  in  antimony  increases,  the  freezing-point 
T  is  lowered;  this  goes  on  until  X  attains  the  value  0.13  and  T  the 
value  228°;  at  this  moment  the  remaining  liquid  solidifies.  Viewed 
with  the  microscope,  the  lingot  obtained  is  seen  to  be  formed  of 
large  lead  crystals  implanted  in  a  finely  grained  eutectic  mixture. 

On  the  contrary,  a  liquid  mixture  for  which  X  is  included 
between  1  and  0.13,  brought  to  the  freezing-point,  deposits  anti- 
mony crystals;  the  proportions  X  in  antimony  decreases,  until  X 
reaches  the  value  0.13  and  T  the  value  228°;  then  the  rest  of  the 
liquid  solidifies  into  a  eutectic  which  cements  the  antimony 
crystals  together,  as  may  be  shown  with  the  microscope. 

According  to  the  same  authors,  the  alloy  formed  of  zinc  (1) 
and  aluminium  (2)  possesses  properties  in  all  points  analogous  to 
the  preceding,  X  increasing  from  0  to  0.05,  the  freezing-point 
decreases  from  T0=433°,  the  zinc  fusing-point,  to  0  =  389°;  the 
solid  produced  is  pure  zinc.  X  continuing  to  increase  from  0.05 
to  1,  the  freezing-point  increases  from  0  =  389°  to  771  =  650°, 
fusing-point  of  aluminium.  The  point 

0=389°,  £=0.05 

is  a  eutedic  point. 

1  H.  GAUTIER,  Bulletin  de   la  Society  d' Encouragement,  Oct.   1896,  and 
Contribution  d  V Etude  des  alliages,  p.  93. 

2  G.  CHARPY,  Butt,  de  la  Soc.  d' Encouragement,  March  1897,  and  Contrib. 
d  VEtude  des  alliages,  pp.  121  and  203. 


MIXED  CRYSTALS.     THE  METALLIC  ALLOYS.          301 

A  case  more  complicated  than  the  preceding  may  occur:  it 
is  that  where  the  liquids  containing  large  proportions  of  the  metal 
1  deposit  the  metal  1  in  the  pure  state,  where  the  liquids  contain- 
ing large  proportions  of  metal  2  deposit  this  metal  in  the  pure  state, 
finally  where  liquids  of  intermediate  composition  deposit  a  definite 
compound. 

The  congelation  curves  have  then  usually  the  shape  we  found 
(Fig.  79,  p.  241)  when  studying  a  mixture  of  two  melted  salts 
where  a  double  salt  may  be  formed. 

The  type  of  these  alloys  seems  to  be  the  alloy  formed  of  tin  (1) 
and  copper  (2),  studied  *  by  H.  Le  Chatelier,  by  Sir  Roberts- Austen 
and  Stansfield,  and  by  G.  Charpy. 

When  X  varies  from  0  to  0.03,  the  solidhying-point  is  lowered 
from  T0=232°,  fusing-point  of  pure  tin,  to  0  =  227°;  the  solid  de- 
posited is  pure  tin. 

When  X  varies  from  0.72  to  1  the  point  of  congelation  rises 
from  W=770°  to  1065°,  fusing-point  of  pure  copper;  the  solid 
deposited  is  pure  copper. 

When  X  increases  from  0.03  to  0.72  the  freezing-points  increase 
constantly  from  6=227°  to  TF=770°;  the  solid  deposited  is  a 
definite  compound:  SnCu3. 

From  what  we  have  just  said,  the  freezing-point  curve  of  this 
definite  compound  does  not  possess  an  indifferent  point;  the  point 

0=227°,  £=0.03 

is  a  eutectic  point;  the  point 

17=770°,  X=0.72 

is  a  transition-point. 

In  other  cases,  the  freezing-point  curve  of  the  definite  com- 
pound has  an  indifferent  point;  the  three  freezing-point  curves 
have  then,  very  exactly,  the  arrangement  shown  in  Fig.  79,  p.  241. 
Such  would  be  the  case  realized,  according  to  Le  Chatelier,2  by  the 
alloys  of  copper  and  antimony  within  which  the  definite  com- 
pound SbCu3  may  be  formed. 

233.  Liquid  metallic  mixtures  which  give  solid  solutions. — 
The  case  we  have  just  mentioned  is  the  simplest,  but  it  appears 

1  See  Contribution  d  V  etude  des  alliages,  pp.  99  and  139. 

3  H.  LE  CHATELIER,  Bulletin  de  la  societe  de  V Encouragement,  1895,  p.  573. 


302 


THERMODYNAMICS  AND  CHEMISTRY. 


to  be  quite  rare;  the  most  often,  when  a  mixture  of  two  fused 
metals  is  cooled,  a  solid  solution  is  obtained  which  contains  the 
two  metals  in  variable  proportion. 

The  simplest  case  to  be  had  is  that  where  two  metals,  isomor- 
phous  with  each  other,  form,  whatever  their  proportions,  a  single 
kind  of  mixed  crystals;  all  the  freezing-points  then  range  them- 
selves along  a  single  curve  (Fig.  86,  p.  261),  joining  the  fusing- 
point  of  one  of  the  metals  to  the  fusing-point  of  the  other. 

This  is  the  case  with  the  alloys  of  gold  and  silver.  The  freez- 
ing-points all  lie  on  a  sensibly  straight  line  extending  from  the 
fusing-point  of  gold  to  that  of  silver. 

The  alloys  of  bismuth  and  antimony,  whose  fusibility  curve 
and  microscopic  structure  have  been  studied  by  Roland-Gosselin 
and  by  Charpy,1  are  of  this  same  type;  the  freezing-points  lie  along 
•a  single  curve  joining  the  fusing-point  TQ  =  2Q8°  of  bismuth  to  the 
fusing-point  771  =  622°  of  antimony. 

The  particularly  simple  case  realized  by  these  alloys  is  quite  rare; 
in  general  there  may  be  formed  two  kinds  of  solid  solutions,  crys- 
tallized or  not ;  the  first,  which  include  as  a  special  case  the  metal  1 
taken  alone,  are  formed  in  liquid  mixtures  rich  in  this  metal;  the 

others,  among  which  should  be  counted 
the  pure  metal  2,  arise  in  liquid  mixtures 
which  enclose  principally  the  metal  2. 

To  these  two  kinds  of  solid  solutions 
correspond  two  distinct  congelation 
curves,  ^  and  c2,  the  first  starting  from 
the  point  Fl  (Fig.  101),  whose  ordinate 
is  the  temperature  of  fusion  of  the 
metal  1,  the  second  starting  from  the 
point  F2,  whose  ordinate  is  the  tempera- 
te ture  of  fusion  of  the  metal  2. 

In  general  the  first  of  these  curves 
descends  from  left   to    right    and    the 

second  from  right  to  left ;  the  arrangement  is  similar  to  that  found 
by  Reinders  in  studying  the  mixtures  in  fusion  of  potassium  ni- 
trate and  sodium  nitrate  (Art.  223). 

The  curves  c  ,  c2  intersect  in  a  certain  point  E  of  coordinates 

B,  e. 

1  G.  CHARPY,  Contribution  d  V etude  des  alliages,  pp.  114  and  138. 


S       £2 
FIG.  101. 


MIXED  CRYSTALS.     THE  METALLIC  ALLOYS.          303 

To  these  curves  must  be  joined  the  fusion  curves  C1;  C2  of  the 
solid  solutions.  To  the  temperature  0  corresponds  on  the  first 
curve  a  point  Av  of  abscissa  £1;  and  on  the  second  curve  a  point 
A2  of  abscissa  £2. 

A  eutectic  conglomerate  of  mean  composition  B  is  produced 
at  the  temperature  6}  this  conglomerate  is  a  juxtaposition  of  masses 
formed  by  the  solid  solution  of  the  first  kind  whose  composition 
is  £lf  and  of  masses  belonging  to  the  solid  solution  of  the  second 
kind  of  composition  £2. 

Their  properties  are  similar  to  those  of  the  alloys  of  silver  and 
copper. 

Solid  alloys  are  known  in  which  the  copper  is  united  to  a  pro- 
portion of  silver  varying  from  0  to  a  certain  limit,  and  alloys  where 
silver  is  united  to  a  proportion  of  copper  variable  from  0  to  a  cer- 
tain limit.  Besides,  an  alloy  is  known,  Levol's  alloy,  in  which 
copper  and  silver  enter  in  a  fixed  ratio.  This  alloy  has  a  definite 
fusing-point,  which  is  0  =  777°.  It  was  for  a  long  time  considered  as 
a  definite  compound,  to  which  the  formula  AggCug  was  attributed. 

In  studying  the  freezing  of  fused  mixtures  of  silver  and  copper, 
Sir  Roberts-Austen1  and  Heycock  and  Neville2  have  determined 
two  congelation  curves  q,  c2;  their  point  of  intersection  has  for  co- 
ordinates exactly  the  composition  and  fusing-point  of  Levol's  alloy; 
the  latter  is  a  eutectic  conglomerate;  by  a  microscopic  exami- 
nation of  Lavol's  alloy  Osmond  3  has  corroborated  this  conclusion. 

The  alloys  of  copper  and  gold  4  give  rise  to  considerations  sim- 
ilar in  all  respects  to  the  preceding. 

2$3a.  Alloys  of  zinc  and  antimony;  Reinders'  researches. — 
The  two  congelation  curves  cl}  c2  of  the  two  kinds  of  mixed  crystals, 
instead  of  uniting  in  a  eutectic  point,  as  happens  in  the  last  two 
cases  we  have  cited,  may  be  joined  in  a  transition-point.  The 
arrangement  they  assume  is  then  similar  to  that  found  by  Hissink 
in  studying  the  congelation  of  silver  nitrate  and  of  sodium  nitrate 
i  Fig.  88,  p.  277). 

A  particularly  remarkable  example,  and  resembling  this  type, 

1  ROBERTS-AUSTEN,   Proceedings  of  the  Royal  Society  of  London,   1875, 
p.  481 ;  Annual  Mint  Report,  1000,  p.  70. 

2  HEYCOCK  and  NEVILLE,  Philosophical  Transactions,  v.  189,  p.  25. 
8  OSMOND,  Comptes  Rendus,  v.  124,  p.  1094,  1897. 

4  ROBERTS-AUSTEN,  Annual  Mint  Report,  1900,  p.  70. 


304  THERMODYNAMICS  AND  CHEMISTRY. 

has  recently  been  studied  by  Reinders;1  it  is  furnished  by  the 
alloys  of  the  two  following  metals:  (1)  zinc,  (2)  antimony. 
Let  MI  and  cu2  be  the  molecular  weights  of  these  two  metals. 

When  —X  varies  from  0  to  1,  the  mixture  may  deposit  four 

OAj 

kinds  of  distinct  mixed  crystals,  which  we  shall  indicate  by  the 
indices  1,  2,  3,  4;  to  these  four  kinds  of  crystals  correspond  four 
different  congelation  curves,  c1;  c2,  c3,  c4;  each  of  these  curves 
is  joined  to  the  following  in  a  transition-point. 

~X  increasing  from  0  to  0.08,  the  freezing-point  rises  along 
w2 

the  curve  c4  from  the  fusing-point  T0=232°  of  pure  tin  to  the  tem- 
perature TF12  =  243°  of  the  first  transition-point. 

— X  increasing  from  0.08  to  0.2,  the  freezing-point  rises  along 

C02 

c.£,  from  W12  to  the  temperature  TF23=310°  of  the  second  transition- 
point. 

— X  increasing  from  0.2  to  0.51,  the  freezing-point  rises  along 
(t)2 
c3,  from  W23  to  TF34  =  430°  of  the  third  transition-point. 

Finally,  — -X  increasing  from  0.51  to  1,  the  freezing-point  rises 
co2 

along  the  curve  c4,  from  W34  up  to  the  point  of  fusion  T  =  622°  of 
pure  antimony. 

In  the  same  way  as  for  the  two  congelation  curves  clt  c2  of 
Fig.  88  (p.  277)  correspond  two  fusion  curves  Clt  C2  joined  to 
each  other  by  a  line  AtA2  parallel  to  OX  and  having  for  constant 
ordinate  the  transition  temperature  W,  so  here  we  shall  have  four 
fusion  curves  Clt  C2,  C8,  C4;  each  of  these  four  curves  will  be  joined 
to  the  next  by  a  straight  f  egment,  parallel  to  OX,  having  for  con- 
stant ordinate  the  ordinate  of  the  corresponding  transition-point. 

According  to  Reinders,  the  first  straight  segment  A±A2,  which 

has  the  constant  ordinate  TF12  =  243,  extends  sensibly  from.— [X= 

0.1  (point  A,)  to  ^X=0.065  (point  AJ. 

The  second  straight  segment  A2A3,  of  constant  ordinate 
TF,3=310°,  runs  sensibly  from  ^-X= 0.3  (point  A2')  to-lZ=0.6 
(point  43). 

1  W.  REINDERS,  Zeitschrift  fur  anorganische  Chemie,  v.  25,  p.  113,  1901. 


MIXED  CRYSTALS.     THE  METALLIC  ALLOYS. 


305 


The  third  straight  segment  A3'A4)  of  constant  ordinate  W^= 430°, 
goes  from  the  point  A3',  whose  abscissa  is  — X=0.55,  to  the  point  A4 

of  abscissa  — .X=0.9. 

The  curves  Clt  C2,  C3,  C4,  whose  extremities  are  thus  known, 
have  not  been  determined. 

2335.  Amalgams  of  cadmium;  Byl's  researches. — Byl 1  has 
lecently  examined  a  system  which  comes  within  the  type  studied 
by  Reinders ;  it  is  the  system  formed  by  mercury  and  cadmium. 

The  congelation  of  liquid  amalgam  may  give  rise,  according 
to  circumstances,  to  two  kinds  of  mixed  crystals. 

The  mixed  crystals  of  the  first  kind,  which  we  shall  call  a 
crystals,  are  isomorphous  with  crystals  of  pure  mercury;  they  are 
deposited  within  liquid  mixtures  containing  a  proportion  of  cad- 
mium less  than  a  certain  limit;  if  we  attribute  the  index  1  to  mer- 
cury and  the  index  2  to  cadmium,  and  if  we  keep  the  notation 

of  the  preceding  article,  this  limit  corresponds  to  —X=0.67. 

The  freezing-point  is  included  between 
—  40°,  freezing-point  of  pure  mercury,  and 
188°.  The  curve  of  congelation  is  the  curve 
AB  (Fig.  D).  The  composition  of  the 
mixed  crystals  obtained  is  comprised 

between   pure  mercury  and  — — X=0.75. 

The  line  of  fusion  for  these  crystals  is 
the  line  AC. 

The  liquid  mixtures  whose  propor- 
tion in  cadmium  exceeds  —X=0.67  give 

other  mixed  crystals,  isomorphous  with 
pure  cadmium,  the  /?  crystals.     As  the 
liquid  becomes   richer   in   cadmium,  the 
freezing-point   rises    from  188°  to   320°, 
freezing-point  of  pure  cadimum,  tracing  A 
the   congelation  line    BD.      The    mixed 
crystals    contain    a   proportion    of    cad-  ° 
mium  which    increases    from    the  lower 


1   x- 


FIG.  D. 


1  H.  C.  BYL,  Zeitschrift  fur  physikalische  Chemie,  v.  41,  p.  641,  1902. 


306  THERMODYNAMICS  AND  CHEMISTRY. 

limit   — LX=0.67  up  to   totality;  the   fusion  line  of  these  crys- 

OJ2 

tals  is  the  line  ED. 

The  a  and  ft  crystals  may  be  transformed  into  each  other. 
When  the  point  representing  the  state  of  the  system  is  included 
between  the  lines  CF  and  EG,  the  system  in  equilibrium  incloses 
the  two  kinds  of  crystals;  if  the  representative  point  is  to  the  "eft 
of  CF,  the  system  is  homogeneous  and  of  form  a;  it  is  homoge- 
neous and  of  form  /?  if  the  representative  point  is  to  the  right  of 
the  line  EG. 

234.  Carburized  iron.  Roozboom's  Theory. — A  considerable 
number  of  alloys  have  been  studied  according  to  the  principle  stated 
above;  this  difficult  study  *  has  given  in  many  cases  results  which 
are  still  but  hypothetical ;  we  shall  not  stop  to  describe  all  the 
results  attained. 

There  are  some,  nevertheless,  which  we  cannot  pass  over  in 
silence;  although  still  incomplete,  they  already  throw  much  light 
on  a  subject  of  the  first  importance;  we  mean  the  researches  which 
concern  the  constitution  of  carburized  iron. 

These  numerous  investigations  have  rendered  it  possible  for 
Roozboom  2  to  give  a  very  atisfactory  epresentation  of  the  phe- 
nomena which  are  produced  within  a  mixture  of  iron  and  carbon 
cooled  with  extreme  slowness  from  the  liquid  state. 

When  the  temperature  of  a  me  ted  mixture  of  iron  and  carbon 
is  lowered  there  are  two  cases  to  consider  according  as  the  amount 
of  carbon  present  in  the  fused  mass  is  less  or  greater  than  4.3%. 

Let  us  give  the  index  1  to  iron  and  the  index  2  to  carbon,  and 

1  See  on  this  subject  ROBERTS- AUSTEN  and  A.  STANSFIELD,  La  Constitu- 
tion des  alliages  metalliques  (Reports  presented  to  the  international  Physics 
Congress,  Paris,  1900,  v.  i,  p.  363).  See  also  Contribution  d  V etude  des 
alliages  metalliques,  loc.  tit. 

3  BAKHUTS-ROOZBOOM,  Eisen  und  Stahl  vom  Standpunkte  der  Phasenkhre, 
Zeitschrift  fur  physikalische  Chemie,  v.  34,  p.  437,  1900;  Iron  and  Steel  from 
the  point  of  view  of  the  "  Phase  Doctrine  "  (Journal  of  the  Iron  and  Steel  Institute, 
No.  2,  1900) ;  STANSFIELD,  The  Present  Position  of  the  Solution  Theory  of  Car- 
burized Iron  (ibid.} ;  BAKHUIS-ROOZBOOM,  Le  Per  et  Vacier  au  point  de  vue  de 
la  doctrine  de  phases  (Contribution,  p.  327);  OSMOND,  Remarques*  sur  U 
memoire  precedent  (same,  p.  370);  H.  LE  CHATELIER,  Observations  sur  le 
memoire  de  M.  Bakhuis-Roozboom  (same,  p.  379). 


MIXED  CRYSTALS.     THE  METALLIC  ALLOYS. 


307 


suppose,  in  the  first  p^ce,  that  the  value  of  x  which  indicates  the 
composition  of  the  casting  is  greater  than  0.043. 

In  these  conditions  the  cooled  casting  deposits  pure  carbon  in 
the  state  of  graphite;  the  temperature  at  which  this  deposit  is 
made,  at  which,  consequently,  the  melted  casting  may  be  regarded 
as  a  saturated  solution  of  graphite  in  iron,  depends  on  the  pro- 
portion of  carbon  in  the  liquid  mixture;  it  is  the  lower  as  #  is 
smaller;  when  x  is  reduced  to  the  value  0.043  this  temperature 
descends  to  1130°.  The  locus  of  points,  which  have  for  abscissae 
values  of  x  and  for  ordinates  the  temperatures,  is  the  curve  c% 
(Fig.  102),  solubility  curve  of  graphite  in  melted  iron. 

T 
1600°  LFi 


0.01       0.02      0.03      0.04       0.05       0-06      0.07  X 

FIG.  102. 

When,  in  the  liquid  mixture,  the  value  of  x  is  less  than  0.043, 
things  happen  quite  otherwise;  by  cooling  the  melted  casting 
mixed  crystals  are  obtained  containing  carbon  and  iron  in  variable 
proportion,  and  to  which  is  given  the  name  martensite;  martensite 
is  the  principal  constituent  of  the  white  casting. 

The  line  q  is  the  congelation  curve  of  martensite  in  the  liquid 
mixture;  it  descends  from  left  to  right  from  the  point  Fl  (z=0, 
!T=16000)  to  the  point  E  (z=0.043,  T=1130°). 

The  study  of  this  congelation  is  not  complete  so  long  as  the 
composition  of  the  martensite  crystals  which  are  formed  at  a  given 


308  THERMODYNAMICS  AND  CHEMISTRY. 

temperature  is  unknown;  in  order  to  know  this  composition,  it 
is  sufficient  to  trace  the  fusion  curve  C^  of  martensite;  starting 
from  the  point  Flf  this  curve  descends  to  the  point  A  of  ordinate 
T=1130°  and  of  abscissa  z=0.02. 

The  point  E  is  a  eutectic  point.  When  the  temperature  is 
lowered  to  this  point  the  liquid  part  of  the  casting  will  certainly 
Jiave  for  composition  x= 0.043. 

By  a  new  lowering  of  temperature,  however  slight,  this  liquid 
will  solidify  and  form  a  eutectic  conglomerate,  containing  on  the 
average  4.3%  carbon;  this  conglomerate  will  be  formed  by  the 
juxtaposition  of  pure  graphite  crystals  and  of  mixed  martensite 
crystals  of  2%  carbon. 

The  solid  conglomerates  furnished  by  freezing  may  undergo,  at 
temperatures  below  1130°,  various  modifications. 

In  the  first  case,  within  the  conglomerates  of  martensite  and 
carbon,  at  a  temperature  below  1000°,  there  may  be  formed  a 
definite  compound,  which  separates  from  the  mass;  amorphous, 
it  forms  a  cement  between  the  crystals  of  martensite  or  of  graphite; 
this  compound,  whose  formula  is  Fe3C,  is  cementite. 

The  system  formed  of  the  two  independent  components  iron, 
and  carbon,  divided  into  three  phases,  graphite,  martensite, 
cementite,  can,  under  atmospheric  pressure,  remain  in  equilibrium 
only  at  a  single  temperature,  the  which  is  in  the  neighborhood  of 
1000°;  the  composition  of  each  of  the  three  phases  in  equilibrium 
is  likewise  determined;  this  condition  is  fulfilled  of  itself  for 
graphite  and  cementite;  the  martensite  crystals  which  may  be 
in  equilibrium  with  these  two  substances  contain  about  1.8% 
carbon  (x =0.018). 

Outside  of  the  conditions  indicated,  one  of  these  phases  will 
disappear  from  the  system. 

If  the  temperature  is  above  1000°,  the  cementite  will  decom- 
pose into  graphite  and  martensite,  which  will  remain  alone  in 
contact  with  each  other. 

When  the  temperature  is  below  1000°,  the  martensite  and  the 
graphite  will  combine  to  give  cementite  until  one  of  the  compo- 
nents has  totally  disappeared. 

Cementite  contains  about  6.6%  carbon  te= 0.066);  if,  there- 
fore the  value  of  x  which  represents  the  average  constitution  of 


MIXED  CRYSTALS.     THE  METALLIC  ALLOYS.  309 

the  conglomerate  exceeds  0.066,  case  in  which  the  representative 
point  will  be  to  the  right  of  the  line  PP'  (x= 0.066),  the  conglom- 
erate having  attained  the  equilibrium  condition  will  contain  only 
cementite  and  graphite;  if,  on  the  contrary,  the  value  of  x  repre- 
senting the  average  composition  of  the  conglomerate  is  less  than 
0.066,  case  for  which  the  representative  point  will  be  to  the  left 
of  the  line  PP*,  the  conglomerate  in  equilibrium  will  be  formed  of 
mixed  crystals  of  martensite  embedded  in  amorphous  cementite 

At  temperatures  included  between  1130°  and  1000°  mixed 
crystals  of  martensite  may  be  observed  in  equilibrium  with  graphite 
crystals;  to  each  temperature  corresponds  an  equilibrium  state 
of  this  bivariant  system  at  atmospheric  pressure  and  the  com- 
position of  each  phase  is  given  for  this  state  of  equilibrium;  there- 
fore at  every  temperature  comprised  between  1130°  and  1000° 
the  martensite  crystals  which  may  coexist  with  the  graphite  crys- 
tals have  a  given  composition,  and  the  law  connecting  this  com- 
position with  the  temperature  may  be  represented  by  a  certain 
curve. 

From  the  meaning  given  to  the  point  A  this  curve  necessarily 
passes  through  this  point;  besides  we  have  seen  that  at  the  tem- 
perature of  1000°  there  corresponds  a  point  B  whose  abscissa  is 
x= 0.018. 

If  the  point  which  represents  the  temperature  and  mean  com- 
position  of  the  system  lies  to  the  left  of  the  line  AB,  there  may 
not  be  established,  within  the  system,  a  state  of  equilibrium  be- 
tween the  martensite  and  the  graphite;  the  martensite  crystals, 
too  poor  in  carbon,  dissolve  the  whole  of  the  graphite. 

At  temperatures  below  1000°  no  further  equilibrium  can  be 
had  between  the  martensite  and  the  graphite,  but  there  may  be 
between  martensite  and  cementite;  the  martensite  crystals  capa- 
ble of  figuring  in  such  an  equilibrium  state  have,  at  each  tempera- 
ture, a  definite  composition  which  corresponds  to  a  point  on  the 
line  Be-,  this  line  starts  necessarily  from  the  point  B  (a; =0.018, 
T = 1000°) .  This  line  descends  to  the  point  e  (x = 0.0085,  T = 690°) , 
whose  importance  we  shall  see  directly. 

Between  1000°  and  690°,  within  systems  too  poor  in  carbon 
to  co-tain  anything  else  than  martensite,  new  transformations 
are  produced;  iron  separates  out  in  the  pure  state;  this  separa- 


310     .  THERMODYNAMICS  AND  CHEMISTRY. 

tion  may  take  place  in  two  different  forms,  which  we  shall  denote 
with  Osmond,  as  Fea  and  Fe/?. 

The  form  Fea  has  a  very  great  magnetic  susceptibility;  this 
quantity  is  very  small  for  the  Fe^  form.  The  Fea  form  changes  into 
the  Fe^  form  when  the  temperature  exceeds  770° ;  on  the  contrary, 
below  this  temperature  iron  passes  from  the  Fe^  form  to  the  Fea 
form. 

A  third  form,  Fer,  is  that  assumed  by  iron  above  890°;  mar- 
tensite  is  isomorphous  with  this  form;  it  may  be  said  that  Fer 
iron  is  martensite  with  0%  carbon. 

The  solubility  curve  of  Fe^  in  martensite  starts  from  the 
point  T  (x=Q,  T  =  890°)  which  corresponds  to  the  transformation- 
point  of  Fe/?  into  Fer  iron;  it  descends  from  left  to  right  to  the 
point  6  (x = 0 . 0035,  T = 770°) . 

From  this  point  6  starts  the  solubility  curve  of  Fea  in  mar- 
tensite, curve  which  descends  to  the  point  e  of  which  we  have 
spoken  above. 

A  system  whose  representative  point  lies  to  the  left  of  the  line 
•cO  is  a  conglomerate  of  Fe^  and  of  martensite  crystals;  a  system 
whose  representative  point  is  to  the  left  of  6e  is  a  conglomerate 
of  Fea  and  martensite. 

The  point  e  is  a  eutectic  point;  the  martensite  capable  of 
existing  as  far  as  this  temperature  (T=690°)  has  a  well-defined 
composition  (#  =  0.0085).  The  least  lowering  of  temperature 
disintegrates  it  and  it  then  forms  a  eutectic  of  the  same  average 
composition  as  formed  by  small  particles  of  ferrite  (Fea)  embedded 
in  cementite.  Arnold  and  Sarby,  who  took  this  eutectic  for  a 
definite  compound,  gave  it  the  name  perlite. 

It  is  clear  that  at  a  temperature  below  690°  a  mixture  of  iron 
and  carbon  in  equilibrium  should  possess  a  state  determined 
solely  by  the  knowledge  of  its  average  composition  x.  Accord- 
ing to  this  value  of  x,  this  state  is  placed  in  one  of  the  three  cate- 
gories which  we  are  going  to  define : 

1°.  If  x  is  included  between  0  and  0.0085,  the  system  is  formed 
of  perlite  (eutectic  of  ferrite  and  cementite)  with  an  excess  of 
ferrite  (Fe0). 

2°.  If  x  is  between  0.0085  and  0.0066,  the  system  consists  of 
perlite  with  an  excess  of  cementite  (Fe3C). 


MIXED  CRYSTALS.     THE  METALLIC  ALLOYS.          311 

3°.  If  x  is  greater  than  0.066,  the  system  is  composed  of 
cementite  and  graphite. 

The  theory  of  carburized  iron,  such  as  we  have  presented  it  as 
developed  by  Roozboom,  cannot  be  regarded  as  definitely  accept- 
able. Stansfield  and  also  Le  Chatelier  have  already  pointed  out 
various  grave  objections  to  the  explanation  by  this  doctrine. 
Other  objections  have  been  raised  recently  by  Charpy  and  Grenet; 1 
further,  these  chemists  have  put  forth  an  opinion  which  merits 
putting  to  the  test.  According  to  them  the  only  state  of  veritable 
equilibrium  which  a  system  composed  of  iron  and  carbon  ma*y 
have  at  low  temperature  would  be  formed  of  the  pure  ferric  in 
contact  with  pure  graphite;  every  other  state  would  be  observable 
only  by  a  phenomenon  of  lag  analogous  to  surfusion.  The  equi- 
librium states,  whose  laws  Roozboom,  Le  Chatelier,  Stansfield, 
and  various  other  chemists  have  tried  to  give,  may  be  compared 
to  the  state  of  equilibrium  observed,  at  44°,  between  liquid  white 
phosphorus  and  solid  white  phosphorus;  then,  these  two  sub- 
stances may  be  transformed  into  red  phosphorus,  the  only  form 
which  is  truly  in  equilibrium  at  low  temperature.  It  is  also  thanks 
to  this  retardation  of  the  transformation  of  white  phosphorus 
into  red  that  Boulouch  was  able  to  observe  the  states  of  equilib- 
rium which  are  produced,  below  100°,  in  the  system  sulphur- 
phosphorus  (see  Art.  2243.). 

For  other  reasons  also,  the  theory  of  Roozboom  cannot  be 
regarded  as  sufficient  to  represent  all  the  properties  of  carburized 
iron. 

We  know  that  a  carburized  iron  possesses,  at  a  given  tempera- 
ture, a  state  which  is  not  determined  by  the  knowledge  of  its  com- 
position alone;  the  permanent  modifications  known  by  the  names 
of  tempering  and  annealing  may  impress  upon  this  state  infinite 
variations;  the  preceding  theory  should  be  regarded,  therefore,  as  a 
simplified  and  ideal  theory,  true  for  perfectly  annealed  systems; 
for  the  systems  which  do  not  enter  into  this  ideal  case  it  must  give 
way  before  a  theory  which  would  be,  doubtless,  of  an  extreme 
complication. 


1  G.  CHARPY  and  L.  GRENET,  Bull  de  la  Society  £ Encouragement  pour 
V Industrie  nationale,  v.  102,  p.  399,  1902. 


CHAPTER  XV. 
CRITICAL  STATES. 

235.  The  critical  point  in  the  vaporization  of  a  single  fluid. — 
Let  us  study  the  vaporization  of  a  fluid,  carbonic  anhydride,  for 
example. 

There  corresponds  to  every  temperature  T  a  pressure  P,  the 
tension  of  saturated  vapor  at  the  temperature  T,  which  assures  the 
equilibrium  between  the  liquid  and  the  vapor. 

When,  at  the  temperature  T  and  under  the  tension  of  satu- 
rated vapor  corresponding  to  this  temperature,  a  unit  mass  of 
vapor  condenses,  there  is  a  liberation  of  a  quantity  of  heat  L, 
which  is  the  heat  of  vaporization  at  the  temperature  T. 

At  the  temperature  T  and  corresponding  vapor  pressure,  unit 
mass  of  vapor  occupies  a  volume  v,  and  unit  mass  of  the  liquid 
occupies  a  volume  i/;  v  and  i/  are  the  specific  volumes  of  the  satu- 
rated vapor  and  of  the  liquid  at  the  temperature  T. 

Let  us  increase  the  temperature  T  and  follow  the  variations 
of  the  four  quantities  P,  L,  v,  v'  whose  definitions  we  have  just 
recalled. 

When  the  temperature  T  increases  to  a  temperature  6,  near 
to  31°.35  C.,  the  tension  of  the  saturated  vapor  increases  to  a  value 
P  near  to  72.9  atmos.  The  heat  of  vaporization  diminishes  and 
approaches  0. 

The  saturated  vapor  becomes  more  and  more  dense,  so  that  its 
specific  volume  v  diminishes;  the  saturated  liquid  becomes  less 
and  less  dense,  so  that  its  specific  volume  increases ;  the  difference 
(v—tf)  approaches  0;  v  and  v'  approach  the  same  value  which 
we  shall  denote  by  U. 

312 


CRITICAL  STATES. 


313 


Thus  at  a  temperature  below  6  but  differing  very  slightly  from 
6  the  saturated  liquid  and  vapor  are  transformed  one  into  the 
other  without  appreciable  absorption  or  liberation  of  heat  and 
without  appreciable  change  of  volume;  the  various  physical 
properties,  optica',  capillary,  etc.,  of  one  of  the  two  phases  cannot 
be  distinguished  from  the  analogous  properties  of  the  other  phase. 

Hence  it  may  be  said  that  when  the  temperature  approaches 
the  value  0,  the  liquid  and  vapor  carbonic  anhydride  approach 
the  same  state,  called  the  critical  state  of  carbonic  anhydride;  6, 
5,  U  are  called  critical  temperature,  critical  p.essure,  critical  vol- 
ume of  this  fluid. 

If,  taking  the  temperatures  for  abscissae  and  the  pressures  for 
ordinates  (Fig.  103)  we  draw  the  curve  C  of  tensions  of  saturated 
vapor,  this  curve  will  rise  from  left  to 
right  to  the  point  r,  of  abscissa  6  and 
ordinate  ^,  which  bears  the  name  critical 
point. 

If  the  temperature  T  exceeds  the 
critical  temperature  0,  it  is  impossible  at 
any  pressure  to  observe  the  carbonic 
anhydride  divided  between  the  two 
liquid  and  vapor  phases;  this  substance 
is  then  constantly  homogeneous  in  a  state 
called  the  gaseous  state. 

At  a  temperature  T,  below  6,  take  the 
system  under  a  pressure  TT  which  is  above 
the  tension 


n 


n 


T        e       T 
PIG.  103. 

of  the  saturated  vapor  at  the  same  temperature;  the 
representative  point  is  then  at  L,  and  the  system  is  a  liquid. 

We  may  cause  the  temperature  and  pressure  to  vary  so 
that  the  representative  point  describes  a  path  such  as 
LMNM'V\  this  path  cuts  the  line  06'  above  the  critical  point 
r,  descends,  remaining  to  the  right  of  66',  again  cuts  this  line  in 
a  point  M'  located  below  the  critical  point  ;-,  and  arrives  at  a 
point  V  whose  abscissa  is  the  initial  temperature  T,  but  whose 
ordinate  a>  is  less  than  the  tension  of  saturated  vapor  for  this 
temperature. 

While  the  representative  point  goes  from  L  to  M  the  whole 
system  is  in  the  liquid  state;  it  is  in  the  gaseous  state  while  the 


314  THERMODYNAMICS  AND  CHEMISTRY. 

representative  point  follows  the  path  MNM'-,  finally,  the  path 
M'V  corresponds  to  a  state  of  homogeneous  vapor. 

Hence,  while  the  representative1  point  describes  such  a  path, 
there  is  observed  at  no  instant  an  abrupt  change  of  any  one  of  the 
properties  of  the  system;  the  passage  from  the  initial  liquid  state 
to  the  final  state  of  vapor  is  made  in  a  gradual  and  perfectly  con- 
tinuous manner. 

One  may  therefore  take  a  system  at  a  temperature  below  the  critical 
temperature,  in  the  form  of  homogeneous  liquid,  and  by  a  gradual 
transformation  exempt  from  every  abrupt  change,  bring  it  back  to 
the  same  temperature  in  the  vapor  form;  it  is  sufficient  to  cause  the 
representative  point  of  the  system  to  describe  a  path  which  goes  around 
the  criti  at  point. 

In  1869  Thomas  Andrews  made  these  very  important  observa- 
tions on  carbonic  anhydride;  since  then  many  observers  have 
repeated  them  on  a  great  number  of  liquids;  the  idea  of  critical 
state  dominates  the  study  of  transformations  of  a  fluid  of  definite 
composition  susceptible  of  existing  in  the  two  states  of  liquid  and 
vapor. 

We  shall  see  that  this  principle  is  susceptible  of  very  great 
extension. 

236.  Double  liquid  mixtures.  The  temperature  at  which  the 
two  layers  have  the  same  composition  does  not  correspond  to  an 
indifferent  point. — With  W.  Alexejew 1  and  V.  Rothmund,2  let  us 
take  a  mixture  of  water  and  phenol ;  the  mixture  of  these  two  liquids 
is  not  always  homogeneous;  when  the  proportions  of  water  and 
phenol  are  properly  chosen,  it  separates  into  two  layers  of  different 
composition  and  densities,  thus  forming  a  double  liquid  mixture 
which  is  a  bivariant  system. 

At  a  given  pressure  and  temperature  each  of  the  two  phases 
in  equilibrium  has  a  definite  composition,  independent  of  the 
masses  of  water  and  phenol  present.  Thus  at  the  temperature 
of  +34°.2  C.  a  gramme  of  the  denser  layer  contains  0.688  gr. 
phenol;  a  gramme  of  the  rarer  layer  contains  0.93  g.  of  the  same 
substance. 


1  W.  ALEXEJEW,  Wiedemann's  Annalen,  p.  28,  v.  305,  1886. 

2  V.  ROTHMUND,  Zeitschrift  fur  physikalische  Chemie,  v.  26,  p.  433,  1898. 


CRITICAL  STATES. 


315 


T 
FIG.  104. 


6      T 


Let  us  denote,  in  general,  by  x  the  number  of  grammes  of 
phenol  contained  in  1  gramme  of  the  rarer  layer,  and  by  X  the 
number  of  grammes  of  phenol  in  1  gramme  of  the  denser  layer; 
X  is  evidently  greater  than  x.  Keeping  the  pressure  constant, 
say  that  of  the  atmosphere,  let  us  study  how  x  and  X  vary  when 
the  temperature  T  varies. 

With  this  object  in  view  take  two  rectangular  axes  (Fig.  104) ; 
lay  off  the  temperatures  T  as  abscissae  and 
the  values  of  x  as  ordinates;  at  a  given 
temperature  T  the  composition  of  one  of 
the  two  layers  is  represented  by  a  point 
M,  of  coordinates  T,  X,  the  composition  of 
the  other  layer  by  a  point  m,  of  coordi- 
dinates  T,  x. 

When  the  temperature  T  varies,  the 
point  M  describes  a  certain  curve  C  and 
the  point  m  another  curve  c. 

When  the  temperature  T,  in  increasing 
tends  towards  a  value  0  =  69°  C.,  the  two 
curves  C  and  c  are  seen  to  approach  each  other,  the  two  points  M 
and  m  approach  a  common  point  F,  the  two  compositions  X  and 
x  approach  the  same  limit  £;  at  the  same  time  that  the  two  curves 
C,  c  join  in  the  point  F,  they  so  meet  each  other  as  to  touch  at 
this  point  a  parallel  to  Ox.  • 

One  might  think  this  phenomenon  comparable  to  those  studied 
in  Chapter  XI,  and  that  the  point  F  is  an  indifferent  point  where 
two  distinct  liquid  layers^  but  of  the  same  composition,  are  in  equi- 
librium with  each  other. 

It  is  easy  to  be  convinced  that  this  supposition  would  be  erro- 
neous. 

If  it  were  exact,  we  might  say  of  the  double  liquid  mixture 
formed  by  water  and  phenol  all  that  has  been  said  of  the  double 
mixture  formed  by  a  mixture  of  volatile  liquids  surmounted  by 
the  mixed  vapor  which  it  gives  off;  one  of  the  two  layers  into  which 
the  double  liquid  mixture  is  separated  would  play  the  same  role 
as  the  mixed  liquid,  the  other  layer  would  play  the  same  role  as 
the  mixed  vapor;  if  X  and  x  were  the  compositions  of  the  two 
layers  capable  of  remaining  in  equilibrium  in  contact  with  each 


316 


THERMODYNAMICS  AND  CHEMISTRY. 


m 


other  at  the  temperature  T,  the  point  M  (Fig.  105)  of  coordinates 
T,  X,  would  describe  a  curve  with  two 
branches  CIC' ;  the  point  m,  of  coordinates 
T,  x,  would  describe  another  curve  of  two 
branches,  clc' ;  these  two  curves  would  pass 
through  the  point  /,  of  coordinates  6,  £, 
and  touch  there  a  parallel  to  Ox;  at  a 
given  temperature  T,  below  6  according  to 
the  masses  of  water  and  phenol  employed, 
our  double  liquid  mixture  could  possess 
two  distinct  states  of  equilibrium;  in  one, 
the  two  superposed  layers  would  have  the 


O  e    T  compositions  X,  x  and  the  representative 


FIG.  105.  points  M,  m ;  in  the  other,  these  two  layers 

would  have  the  compositions  X' ',  x'  and  the  representative  points 
M',  m'. 

This  disposition  is  in  conformity  with  that  shown  in  Fig.  61 
for  a  mixture  of  volatile  liquids  whose  boiling  point  passes  through 
a  maximum  value;  it  has  not,  on  the  contrary,  any  analogy  with 
that  shown  in  Fig.  104.  When,  therefore,  the  temperature  ap- 
proaches 69°,  we  cannot  suppose  that  the  two  layers  into  which 
the  mixture  of  water  and  phenol  is  divided  approach  two  distinct 
solutions  having  the  same  composition;  and  as  there  is  no  doubt 
that  they  tend  to  have  the  same  composition,  we  are  constrained 
to  admit  that  they  have  as  limiting  state,  not  two  distinct  solu- 
tions, but  a  single  solution. 

237.  This  temperature  is  a  critical  temperature. — We  may 
therefore  formulate  the  following  proposition: 

When  the  temperature  of  a  double  liquid  mix  ure  is  gradually 
raised,  it  may  happen  that  the  composition  and  the  various  properties 
of  the  two  layers  into  which  it  is  divided  differ  less  and  less;  when 
the  temperature  reaches  a  certain  value  6  the  two  layers  become  identi- 
cal in  all  respects. 

The  analogy  between  this  law  and  that  discovered  by  Andrews 
in  the  study  of  the  vaporization  and  liquefaction  of  a  fluid  of  defi- 
nite composition  is  evident;  in  virtue  of  this  analogy,  we  shall 
say  that  0  is  the  critical  temperature  and  £  the  critical  composition 
oj  the  liquid  mixture  studied,  under  the  pressure  considered. 


CRITICAL  STATES. 


317 


At  a  temperature  below  the  critical  temperature  the  mixture 
is  divided  into  two  distinct  layers,  separated  by  a  quite  sharp 
surface  of  contact;  when  the  temperature  reaches  the  critical  value 
the  surface  of  separation  becomes  indistinct  and  disappears  and, 
instead  of  two  separate  layers,  there  remains,  at  temperatures 
above  6,  but  one  homogeneous  mixture. 

At  a  temperature  T,  lower  than  the  critical  temperature  0,  the 
mixture  is  in  a  homogeneous  state  which  we  shall  call  upper  layer, 
if  it  is  rich  enough  in  water  for  the  representative  point  S  (Fig.  106) 
to  lie  below  the  curve  c;  it  is  divided 
into  two  layers  if  the  representative 
point  is  between  the  curves  c  and  C; 
it  is  in  the  homogeneous  state  which 
we  shall  call  lower  layer,  if  it  is  rich 
enough  in  phenol  for  the  representa- 
tive point  to  be  at  /,  above  the  curve  C. 

Take  the  system  in  the  last  state; 
raise  the  temperature  and  add  water, 
so  that  the  representative  point  de- 
scribes the  path  IMN,  which  cuts  the 
line  00'  in  M,  above  the  critical  point  o 
F't  then  continue  to  add  water,  but 
lower  the  temperature  so  that  the  representative  point  describes 
the  path  NM'S  which  cuts  the  line  00'  at  M',  below  the  critical 
point  F;  the  system,  taken  at  the  temperature  T  and  in  the  state 
of  upper  layer,  returns  to  the  same  temperature,  but  in  the  state 
of  lower  layer;  during  the  modification,  it  has  remained  homo- 
geneous and  has  undergone  no  sudden  change,  but  its  various 
properties  have  altered  gradually. 

A  liquid  mixture  may  therefore  pass  from  the  state  of  upper  layer 
to  the  state  of  lower  layer  by  a  continuous  ransformation;  it  suffices 
that  the  path  of  the  representative  point  goes  around  the  critical 
point. 

238.  Mixtures  which  separate  into  two  layers  at  tempera- 
tures below  the  critical  temperature. — The  water-phenol  mixture 
is  not  the  only  one  which  may  separate  into  two  layers  below  a 
certain  critical  temperature  0  and  which  remains  perforce  homo- 
geneous at  a  temperature  above  0;  the  existence  of  such  a  crit~ 


T  e 

FIG.  106. 


318 


THERMODYNAMICS  AND  CHEMISTRY. 


ical  temperature  is  a  very  general  phenomenon,  as  may  be  seen 
from  the  following  table. 


Mixtures  studied. 

0 

Observers. 

Phenol-water  

+  69° 

W.  Alexejew 

Benzole  acid-water  

116° 

Phenolate  of  phenylammonium-water  .  . 
Aniline—  water  . 

140° 
166° 

<c 
u 

Secondary  butyl  aclohol—  water 

108° 

tt 

Isobutyl-alcohol—  water 

132° 

ti 

Proprionitrite—  water  ... 

113° 

V  Rothmund 

Salicylic  acid—  water 

95° 

Furfurol—  water 

122° 

Ace  ty  lace  tone—  water  

88° 

Isobutric  acid—  water  ... 

24° 

Methylethylcetone—  water     

151° 

Succinonitrite—  water.  .           

55° 

Schreinemaker  l 

Chlorobenzine-sulphur.              

116° 

W.  Alexejew 

Essence  of  mustard—  sulphur  

124° 

Aniline—  sulphur  

138° 

i 

Benzine—  sulphur  

164° 

< 

Toluene—  sulphur  .  .    .  . 

179° 

e 

Sulphide  of  carbon—  methyl  alcohol  .  .    .  . 

40° 

V.  Rothmund 

Hexane—  methyl  alcohol  

42° 

u 

Resorcine-benzine  

109°    • 

u 

Zinc-bismuth  -I 

Between  800° 

W.  Spring  and 

and      900° 

Romanow  2 

239.  Mixtures  which  separate  into  two  layers  at  tempera- 
tures higher  than  the  critical  point. — In  all  the  cases  we  have  just 
cited,  the  mixture  susceptible  of  being  divided  into  two  layers 
at  temperatures  below  the  critical  point  6  is  of  necessity 
homogeneous  at  temperatures  higher  than  6;  we  possess  like- 
wise examples  (see  accompanying  table),  less  numerous  it  is 


Mixtures  studied. 

e 

Observers. 

Die  thylamine—  water 

+  122° 

F.  Guthrie  3 

/?-Collidine—  water 

4° 

V.  Rothmund 

Triethylamine—  water                  .            .  . 

20° 

tt 

1  SCHREINEMAKERS,  Zeitschrift  fur  physikalische  Chemie,  v.  23,  p.  417,  1897. 
3  SPRING  and  ROMANOW,  Zeitschrift  fur  anorganische  Chemie,  v.  13,  p.  29, 


1897. 


3  F.  GUTHRIE,  Philosophical  Magazine,  5th  S.,  v.  18,  pp.  29  and  499,  1884. 


CRITICAL  STATES. 


319 


true,  of  mixtures  which  remain  homogeneous  at  temperatures 
below  the  critical  temperature  0,  while 
at  temperatures  above  0  it  may  sepa- 
rate into  two  layers;  the  two  curves 
C  and  c  are  then  arranged  as  shown  in 
Fig.  107. 

240.  Influence   of  pressure  on  the 
critical  temperature  of  a  double  liquid 
mixture. — The    critical    temperature  0 
and  the  critical  pressure  £  may  naturally 
depend    upon    the    pressure    at   which 
the  mixture  is  studied.     Van  der  Lee  l 
has   found    that    the    critical    temper- 
ature of  a  water-phenol  mixture  rises  at  the  same  time  as  the 
pressure 

Further,  it  is  necessary  to  impose  upon  the  pressure  a  very 
considerable  increase  in  order  to  obtain  an  appreciable  variation 
of  the  critical  point. 

241.  Vaporization  of  a  mixture  of  two  liquids;   critical  line; 
dew  surface,  ebullition  surface. — It  is  the  same  when  the  two  phases 
into  which  the  liquid  mixture  is  divided  are  a  liquid  phase  (lower 

layer)  and  a  vapor  phase  (upper  layer) ;  in 
this  case  every  variation  of  the  pressure 
imposes  variations  of  the  same  order  of 
magnitude  on  the  critical  temperature 
and  on  the  critical  composition  of  a  mix- 
ture of  given  composition. 

If,  therefore,  we  lay  off  on  three  axes 
of  rectangular  coordinates  the  tempera- 
ture T  (Fig.  108),  the  pressure  TT,  the 
composition  x,  to  every  pressure  n 
will  correspond  a  point  F  whose  coordi- 
nates  will  give  the  critical  t  mperature 
FIG  108  and  critical  composition  w  th  respect  to 

this    pressure.     When    the    pressure    n 
varies,  thi    point  describes  a  line,  the  critical  line  FT'. 


1  VAN  DER  LEE,  Academy  of  Sciences  of  Amsterdam,  Oct.  29,  1898. 


320  THERMODYNAMICS  AND  CHEMISTRY. 

In  varying  the  pressure,  it  is  not  merely  the  elements  of 
the  critical  point  which  are  varied;  each  of  the  two  curves  C,  c 
is  displaced  at  the  same  tune,  or,  better,  the  continuous  curve 
cC  which  they  form  together.  This  curve  will  generate  a 
certain  surface  which  we  shall  call  the  limiting  surface.  The 
limiting  surface  will  be  composed  of  two  sheets,  the  one  s 
generated  by  the  curve  c,  and  the  other  S  generated  by  the  curve 
C;  we  shall  call  the  sheet  s  the  dew  surface  and  the  sheet  S  the 
ebullition  surface;  these  definitions  will  be  justified  shortly. 

These  two  sheets  meet  each  other  along  the  critical  line.  If 
the  limiting  surface  is  projected  on  the  plane  TOn,  it  is  clear  that 
the  projection  ff  of  the  critical  line  will  make  a  part  of  the  con- 
tour of  the  projection. 

Through  the  point  M,  taken  in  the  plane  TOn,  draw  a  parallel 
to  Ox;  it  will  meet  the  dew  surface  s  at  R  and  the  ebullition  surface 
S&tE. 

Choose  a  starting-point  M  located  above  E  and,  without  chang- 
ing pressure  it  or  temperature  T,  add  to  the  system  a  gradually 
increasing  mass  of  the  fluid  1;  x  will  diminish  and  the  point  M 
will  describe  the  line  Mm. 

As  long  as  the  representative  point  lies  above  E,  the  system 
will  remain  in  the  state  of  homogeneous  liquid;  the  instant  the 
representative  point  reaches  the  position  E}  the  second  phase 
will  appear  in  the  system  under  the  form  of  a  bubble  of  vapor; 
there  will  be  ebullition. 

The  representative  point  lying  between  E  and  R,  the  system 
will  be  divided  into  two  phases;  the  mixed  liquid  will  be  sur- 
mounted by  a  layer  of  vapor.  When  the  representative  point 
reaches  R,  the  last  drop  of  liquid  will  disappear ;  if  the  representa- 
tive point  continued  to  descend,  the  system  would  be  in  the  state 
of  homogeneous  vapor. 

If,  at  this  moment,  we  introduced  some  of  fluid  2  to  increase  x, 
the  representative  point  would  remount;  the  system  would  be  at 
first  in  the  state  of  homogeneous  vapor,  but  at  the  moment  the 
representative  point  attained  the  position  R,  a  liquid  drop  would 
appear;  there  would  be  a  deposit  of  dew. 

To  sum  up,  in  order  that  the  system  may  remain  in  equilibrium 
in  a  state  where  it  is  divided  into  two  phases,  liquid  and  vapor, 


CRITICAL  STATES. 


321 


it  is  necessary  that  the  representative  point  lie  between  the  two 
sheets  s,  S  of  the  limiting  surface;  if  it  leaves  this  region  by 
traversing  the  surface  S,  the  system  passes  into  the  state  of  homo- 
geneous liquid;  if  it  leaves  through  the  surface  s,  the  system 
vaporizes  entirely. 

242.  Dew  line  and  ebullition  line  of  a  compound  of  given 
composition. — When  it  is  desired  to  study  the  vaporization  or 
condensation  of  a  fluid  mixture,  it  is  enclosed  in  a  Cailletet  tube  and 
the  temperature  and  pressure  are  varied;  during  these  operations 
the  value  of  x  which  characterizes  the  average  composition  of  the 
mixture  rests  invariable.  It  is  therefore  interesting  to  discuss  the 
properties  of  a  system  for  which  is  given  the  value  x  of  the  average 
composition. 

The  plane  perpendicular  to  Ox  which  corresponds  to  this  value 
of  x  cuts  (Fig.  109)  the  ebullition  surface  S  in  the  curve  &  and  the 


FIG.  109. 

dew  surface  in  the  curve  (R;  these  two  curves  meet  in  the  critical 
point  F  of  the  mixture  of  concentration  x}  so  as  to  form  a  single 
curve,  section  of  the  limiting  surface  made  by  the  plane  con- 
sidered. 

For  the  particular  case  when  x=0  the  two  lines  &  and  (R 
coalesce  into  a  single  line  Vlt  which  is  the  curve  of  tensions  of 


322  THERMODYNAMICS  AND  CHEMISTRY. 

saturated  vapor  of  fluid  1  in  the  pure  state;  this  line  ends  at  the 
point  Cl}  critical  point  of  the  fluid  1. 

Similarly,  for  the  case  when  x=l  the  two  lines  £  and  (R  be- 
come a  single  line  F2,  which  is  the  curve  of  tensions  of  saturated 
vapor  of  fluid  2  in  the  pure  state;  this  line  ends  at  the  critical  point 
C2  of  the  fluid  2. 

The  critical  line  CJTC2  unites  the  point  Cl  to  the  point  C2. 

Project  the  figure  on  the  plane  TO  • ,  where  x  =  0. 

The  curve  V2  of  tensions  of  saturated  vapor  of  the  fluid  2  is 
projected  in  its  true  magnitude  along  the  line  v2,  which  ends  at 
the  point  c2,  projection  of  the  point  C2. 

The  critical  line  is  projected  along  the  line  C^c,;  this  latter 
is  a  part  of  the  contour  of  the  projection  of  the  limiting  surface. 

The  line  &T61  is  projected  in  its  true  magnitude  along  the  line 
£77),  which  is  the  limiting  line  of  the  mixture  of  composition  x ;  e  is  its 
ebullition  line,  and  p  the  dew  line;  they  meet  in  the  point  7-,  pro- 
jection of  the  critical  point  F,  and  at  this  point  they  are  tangent  to 
the  projection  Ctfc2  of  the  critical  line. 

Take  a  mixture  of  composition  xt  at  a  temperature  T  and  under 
a  pressure  n  which  serve  as  coordinates  to  a  point  in  the  plane 
TOx.  When  this  representative  point  is  in  the  interior  of  the 
limiting  curve  £fp,  the  mixture  of  mean  composition  x  is  divided 
into  two  phases,  a  mixed  liquid  and  a  mixed  vapor.  One  of  these 
two  phases  disappears  and  the  system  becomes  homogeneous 
when  the  representative  point  passes  beyond  the  limiting  line. 
It  is  the  vapor  phase  which  disappears  if  the  representative  point 
passes  beyond  the  limiting  line  at  a  point  which  belongs  to  the 
ebullition  line;  it  is,  on  the  contrary,  the  liquid  phase  which  dis- 
appears if  the  representative  point  passes  beyond  the  limiting  line 
at  a  point  belonging  to  the  dew  line. 

243.  Normal  condensation.  Retrograde  condensation. — The 
consideration  of  limiting  lines  plays  an  .important  role  in  all  the 
researches  relative  to  the  liquefaction  and  vaporization  of  fluid 
mixtures.  The  detailed  analysis  of  these  researches  would  ex- 
ceed the  plan  of  this  work;  so  we  shall  not  give  it.  We  shall  be 
content  to  notice  a  remarkable  consequence  of  the  preceding 
theories. 

Suppose  that  the  disposition  of  the  limiting  line  be  that  repre- 


CRITICAL  STATES. 


323 


sented  in  Fig.  110;  the  point  M,  whose  abscissa  T  is  a  maximum, 
belongs  to  the  dew-line;  experiment  shows  this  to  be  so  in  a  great 
number  of  cases. 

Take  first  a  temperature  T  less  than  the  critical  temperature 
6  of  the  mixture  of  composition  x\  at  this  temperature  cause  the 
pressure  to  increase  gradually  from  a  very  low  value  to  a  very 
great  value;  the  representative  point  will  rise  constantly  along 
the  straight  line  TT',  which  meets  the  dew-line  in  a  point  R  and 
the  ebullition-line  in  a  higher  point  E. 

So  long  as  the  pressure  has  not  reached  the  value  TR  the 
system  will  be  in  a  state  of  homogeneous  vapor;  the  instant  the 


T 

FIG.  110. 


e   t 


pressure  attains  this  value  which  corresponds  to  the  dew-point,  a 
first  drop  of  liquid  will  appear;  the  pressure  continuing  to  in- 
crease, the  mass  of  liquid  will  increase  at  the  expense  of  the  vapor; 
when  the  pressure  reaches  the  value  TE,  which  corresponds  to  the 
ebuUition-point,  the  last  bubble  of  vapor  will  disappear,  and  for 
higher  pressures  the  system  will  be  in  a  state  of  homogeneous 
liquid.  If  the  pressure  be  decreased,  the  same  phenomena  will  be 
produced  in  the  inverse  ord  r. 

When  one  observes  the  succession  of  phenomena  which  we 
have  just  enumerated,  the  system  is  said  to  undergo  normal  con- 
densation. 

The  succession  of  observed  facts  is  quite  otherwise  when  the 
system  is  compressed  keepi  g  corstant  a  temperature  t,  higher 


324  THERMODYNAMICS  AND  CHEMISTRY. 

than  the  critical  temperature  6,  and  also  less  than  the  tempera- 
ture T. 

The  representative  point  rises  along  the  line  #',  which  cuts  the 
dew-line  in  a  first  point,  Io1,  then  in  a  second  point,  p2,  of  ordinate 
greater  than  the  first. 

While  the  pressure  is  less  than  tp^  the  system  remains  in  a  state 
of  homogeneous  vapor;  at  the  instant  the  pressure  reaches  the 
value  tpl}  which  corresponds  to  the  first  dew-point,  a  liquid  drop 
appears;  the  pressure  continuing  to  increase,  the  mass  of  liquid 
increases  at  first,  but  afterwards  passes  through  a  maximum,  then 
diminishes,  and  at  the  instant  the  pressure  attains  tLe  value  tp2, 
which  corresponds  to  the  second  dew-point,  the  last  drop  of  liquid 
disappears;  if  the  system  is  compressed  further,  it  remains  in  the 
state  of  homogeneous  vapor. 

This  series  of  phenomena  constitutes  retrograde  condensation. 

Retrograde  condensation  was  discovered  in  1880  by  L.  Cailletet, 
who  was  studying  the  liquefaction  of  a  mixture  of  air  and  carbonic 
acid  gas;  the  following  year  Van  der  Waals  independently  made 
this  observation  which  he  thought  new;  confirmed  by  the  re- 
searches of  Cailletet  and  Hautefeuille,  and  of  Andrews,  the  phe- 
nomenon of  retrograde  condensation  plays  a  great  part  in  the 
theoretical  and  experimental  studies  relative  to  the  liquefaction 
of  gaseous  mixtures;  these  studies,  developed  by  numerous  physi- 
cists, among  whom  we  may  mention  Van  der  Waals,  Kuenen,  and 
Caubet,  cannot  be  analyzed  here;  the  interested  reader  of  this 
important  question  can  refer  to  Caubet's  treatise.1 

244.  Critical  states  in  the  mixtures  of  three  substances. — A 
great  number  of  liquid  mixtures  formed  by  the  reunion  of  three 
substances  are  susceptible  of  separating  themselves  into  two 
layers,  of  different  composition,  which  remain  in  equilibrium  in 
contact  with  each  other;  in  certain  conditions,  these  two  layers 
take  on  the  same  composition,  the  same  density,  the  same  physi- 
cal properties;  in  other  terms,  when  these  conditi  ns  are  nearly 
realized  the  two  distinct  phases  into  which  the  system  is  divided 
tend  towards  a  common  limiting  state  which  is  a  critical  state. 


1  F.  CAUBET,  Liquefaction  des  melanges  gaseux  (Mem.  de  la  Soc.  physiques 
et  naturelles  de  Bordeaux,  5th  Series,  v.  i,  1901),  and  A.  Hermann,  Paris,  1901. 


CRITICAL  STATES.  325 

The  existence  of  a  critical  state  in  the  mixtures  formed  of  three 
liquid  substances  was  noticed  in  1876  by  Duclaux  when  studying 
the  following  mixtures: 

Amyl  alcohol-alcohol-water; 
Alcohol-ether-water ; 
Acetic  acid-ether-water. 

For  some  years  past,  the  study  of  ternary  liquid  mixtures  and 
their  critical  states  has  been  the  object  of  numerous  important  in- 
vestigations, both  theoretical  and  experimental,  due  to  Schreine- 
makers,1  Snell,2  and  G.  Bruni; 3  we  must  limit  ourselves  here  to  the 
mentioning  of  these  investigations. 

245.  Limiting  crystalline  forms. — The  existence  of  critical 
states  appears,  therefore,  to  be  very  general;  it  is  possible  to  verify 
it  in  the  majority  of  cases  where  a  fluid  system  is  divided  into  two 
phases,  whether  the  system  be  formed  of  one,  two,  or  three  com-, 
ponents;  whether  the  two  phases  be  liquid,  or  one  a  liquid  and 
the  other  a  vapor. 

It  is  probable  also  that  the  notion  of  critical  state  should  not 
be  restricted  to  fluid  systems. 

Pasteur  noticed  that  the  two  crystalline  forms  of  a  dimorphous 
substance  are,  in  general,  little  different  from  each  other;  when 
the  temperature  and  pressure  are  increased,  these  forms  are  modi- 
fied in  such  a  way  that  the  characteristics  which  distinguish  them 
are  attenuated;  we  may  imagine  that,  the  temperature  and 
pressure  *  tending  towards  certain  well-determined  values,  which 
would  be  the  critical  temperature  and  the  critical  pressure,  the  two 
crystalline  forms  would  approach  a  common  limiting  form,  which 
would  be  the  critical  form;  the  dimorphism  of  a  substance  would 
then  be  comparable  to  the  coexistence  in  a  fluid  of  the  two 
forms  liquid  and  vapor,  the  limiting  crystaDine  form  replacing 

1  SCHREINEMAKERS,  numerous  memoirs  published  since  1897,  in  the 
Archives  neerlandaises  des  Societes  exactes  et  naturelles  and  in  the  Zeitschnft 
jur  physikalische  Chemie. 

SNELL,  Journal  of  Physical  Chemistry,  v.  2,  p.  457,  1898. 

8  G.  BRDNI,  Rendiconti  dell'  Accademia  dei  Lincei,  5th  Series,  v.  8,  p.  141, 
1899. 

4  To  be  exact,  it  would  be  necessary  to  speak  here  not  of  the  pressure,  but 
of  the  six  components  of  the  elastic  actions. 


326  THERMODYNAMICS  AND  CHEMISTRY. 

the  gaseous  state,  common  limit  of  the  liquid  state  and  of  the 
vapor  state.  • 

Two  isodimorphous  substances  may  furnish  two  kinds  of  mixed 
crystals;  the  crystalline  forms  of  these  two  kinds  of  crystals  are. 
in  general,  but  slightly  different.  Thus,  as  was  shown  by  Retgers 
(Art.  219),  one  may  obtain  mixed  crystals  of  calcic  carbonate  and 
of  magnesium  carbonate  which  are  isomorphous  with  calcite; 
some  may  be  obtained  which  are  isomorphous  with  magnesite. 
Now,  the  primitive  form  of  calcite  and  the  primitive  form  of  mag- 
nesite are  two  rhombohedrons  having  slightly  different  angles; 
these  angles,  further,  vary  with  the  temperature  and  the  elastic 
actions.  One  may  suppose,  therefore,  that  mixed  critical  crystals 
may  exist  possessing  a  common  limit  for  the  two  forms  of  mixed 
crystals  which  are  observed  under  ordinary  conditions.  These 
limiting  mixed  crystals  would  be  comparable  to  the  critical  states 
in  which  the  two  phases  of  a  mixture  of  two  liquids  blend. 


CHAPTER  XVI. 
CHEMICAL    MECHANICS   OF    PERFECT    GASES. 

246.  Necessity  of  new  hypotheses  in  order  to  penetrate  further 
into  the  study  of  chemical  systems. — All  we  have  said  as  yet  on 
the  subject  of  the  various  problems  of  chemical  mechanics  is  of 
very  great  generality;  a  single  hypothesis,  which  neglects  the 
capillary  actions,  was  used  to  specify  the  form  of  the  internal 
potential  of  the  systems  studied  (see  Chapter  VI,  Art.  89). 

This  great  generality,  which  gives  value  to  the  considerations 
developed  in  the  preceding  chapters,  is  not  without  some  incon- 
venience; from  the  fact  that  they  are  of  wide  extent,  the  princi- 
ples set  forth  are  less  able  to  aid  in  penetrating  into  the  detail  of 
phenomena.  We  know,  for  instance,  that,  under  a  given  pressure, 
it  suffices  for  the  temperature  to  be  determined  in  order  that  the 
saturated  solution  of  a  certain  salt  may  have  a  definite  concen- 
tration; we  know  that  this  concentration  varies  in  the  same  sense 
as  the  temperature  or  hi  the  opposite  sense,  according  as  the  salt 
is  dissolved,  in  saturated  solution,  with  absorption  or  liberation 
of  heat;  but  there  our  information  ceases;  now,  it  is  clear  this  is 
far  from  being  complete  and  we  may  legitimately  desire  more; 
we  may,  for  example,  ask  to  know,  hi  an  exact  or  approximate 
way,  the  form  of  the  law  which  connects  the  concentration  of  the 
saturated  solution  with  the  temperature. 

But  to  obtain,  by  means  of  thermodynamical  principles,  more 
detailed  propositions  than  those  we  have  as  yet  secured,  it  is 
necessary  to  join  to  the  hypothesis  we  have  already  introduced 
new  hypotheses  of  a  more  particular  nature ;  the  difficulty  consists 
in  discovering  such  hypotheses  as  will  furnish  in  cases  of  a  sufficient 
generality  exact  consequences  or,  at  least,  those  of  a  sufficient 
approximation. 

327 


328  THERMODYNAMICS  AND  CHEMISTRY. 

247.  Properties   of   the   systems  to  be  studied. — Gibbs   and 
Horstmann  have  succeeded  in  denning  such  cases  and  in  formu- 
lating such  hypotheses. 

The  chemical  systems  to  which  the  theory  developed  by  these 
physicists  is  applied  may  contain  solids,  liquids,  and  gases. 

The  solids  and  the  liquids  do  not  mix  with  each  other  and  do 
not  dissolve  the  gases,  so  that  each  of  the  solid  or  liquid  phases 
which  the  system  contains  is  a  definite  and  pure  chemical  compound. 

The  specific  volume  of  each  of  the  solid  or  liquid  phases  is  neg- 
ligible with  respect  to  the  specific  volumes  of  the  gases  considered. 

The  specific  heat  of  each  of  these  solid  or  liquid  substances  is 
sensibly  independent  of  the  temperature. 

The  gases  contained  in  the  system  are  in  the  perfect  state. 

248.  Hypotheses   which   characterize   a   mixture   of  perfect 
gases. — These  various  suppositions  would  suffice  to  put  into  equa- 
tions in  a  complete  manner  the  many  problems  of  chemical  me- 
chanics if  the  gaseous  phase  which  the  system  is  supposed  to 
contain  was  formed  by  a  single  gas;  but  in  a  great  number  of  im- 
portant cases  this  phase  is  a  mixture  of  several  gases  which  must, 
according  to  the  principles  just  stated,  be  regarded  as  perfect  gases; 
we  are  therefore  led  to  ask  ourselves  the  following  question:  From 
the  view-point  of  thermodynamics,  how  will  the  mixture  of  two 
or  more  perfect  gases  be  characterized? 

There  is  a  certain  number  of  propositions  which  all  physicists 
are  agreed  to  regard  as  characterizing  the  mixtures  of  two  or  more 
perfect  gases;  let  us  recall  these  propositions. 

The  first  is  this:  A  mixture  of  two  or  more  perfect  gases,  taken 
in  known  proportions,  acts  in  all  circumstances  like  a  single  perfect 
gas.  Thus,  for  example,  air,  which  for  the  chemists  is  a  mixture  of 
several  gases,  is  constantly  cited  and  studied  by  physicists  as  type 
of  a  gas  near  to  the  perfect  state. 

The  second  is  the  one  discovered  by  Berthollet  and  which  is 
known  under  the  name  of  law  of  the  mixture  of  gases:  To  keep  in 
equilibrium  at  a  given  volume  and  pressure  a  mixture  of  perfect  gases, 
it  is  necessary  to  submit  it  to  a  pressure  equal  to  the  sum  of  the  pres- 
sures which  would  be  maintained  separately,  at  the  same  volume  and 
temperature,  by  each  of  the  mixed  gases. 

The  third  may  be  stated  as  follows : 


CHEMICAL  MECHANICS  OF  PERFECT  GASES.  329 

//  two  vessels,  which  contain  at  the  same  pressure  and  temperature 
two  different  perfect  gases  susceptible  of  mixing,  are  put  into  commu- 
nication with  each  other,  the  two  gases  diffuse  into  one  another  with- 
out absorbing  or  giving  out  heat. 

Finally,  the  fourth  proposition  is  the  classic  law  of  the  mixture 
of  gases  and  vapors,  well  known  in  the  following  form : 

When  a  liquid  is  in  equilibrium,  at  a  given  temperature,  with 
its  vapor  mixed  with  a  gas,  the  tension  of  the  gaseous  mixture  is  the 
sum  of  the  tensions  which  is  attained,  at  the  same  temperature,  by  the 
saturated  vapor  of  the  same  liquid1  in  a  space  previously  empty  and 
of  the  pressure  which  would  be  maintained  by  the  gas,  at  the  same 
temperature,  of  volume  equal  to  the  volume  of  the  mixture. 

These  several  laws  completely  characterize,  from  the  thermo- 
dynamic  point  of  view,  the  properties  of  a  mixture  of  perfect 
gases;  they  lead  in  fact  to  the  following  proposition,  which  allows 
calculating  all  these  properties  when  those  of  mixed  gases  are 
known: 

The  internal  potential  of  a  mixture  of  perfect  gases  is  always  equal 
to  the  sum  of  the  internal  potentials  which  it  would  be  proper  to  attribute 
to  each  of  the  mixed  gases  if  it  occupied  alone,  at  the  same  tempera- 
ture, the  entire  volume  of  the  mixture. 

249.  Notations. — We  know  now  the  source  of  Gibbs'  and  of 
Horstmann's  theory;  let  us  leave  the  steps  in  the  development 
of  this  theory  and  state  the  succession  of  results  to  which  it  leads. 

These  consequences  are  condensed  into  three  essential  formulae. 

Imagine  a  system  in  which  either  a  definite  chemical  reaction 
or  an  inverse  reaction  may  be  produced ;  this  will  be,  for  instance, 
a  system  containing  hydrogen,  silver,  hydrogen  sulphide,  silver 
sulphide ;  the  reaction 

Ag2S  +  H2=H2S+Ag2 

may  be  produced  or  the  inverse  reaction. 

Let  us  write,  as  we  have  just  done  for  this  example,  the  chemical 
equation  which  represents  the  former  reaction.  In  the  first  mem- 
ber figure  certain  gaseous  substances  a1}  a?,  .  .  .  and  also  certain 
solids  or  liquids  Alt  A 2,  .  .  .;  in  the  second  member  appear  certain 
gaseous  substances  a/,  a/,  .  .  .  and  certain  solid  or  liquid  bodies 


330 


THERMODYNAMICS  AND  CHEMISTRY. 


Let  us  denote  in  the  following  manner  the  molecular  weights 
and  the  numbers  of  molecules  of  these  various  substances  which 
figure  in  the  chemical  equation: 


Substance. 

Molecular 
Weight. 

Number  of 
Reacting 
Molecules. 

2 

£ 

«; 

1; 

JCj 

% 

«/ 

a/ 

< 

•/ 

AS 

A,' 

*.' 
*>' 

^ 

The  first  member  of  our  chemical  equation  represents  a  mass 


The  second  member  represents  a  mass  : 


These  two  masses  are  equal  to  each  other.     Let  P  be  their 
common  value: 


(1) 


Let  ffj,  <72,  .  .  .  a/,  a/,  ...  be  the  volumes  occupied,  in  the 
normal  conditions  of  temperature  and  pressure,  by  1  gramme  of 
each  of  the  gases  alf  a2,  .  .  .  a/,  a2',  .  .  .;  the  masses  of  these 
gases  which  figure  in  the  chemical  equation  occupy  respectively, 
in  the  normal  conditions  of  temperature  and  pressure,  the  volumes 


CHEMICAL  MECHANICS  OF  PERFECT  GASES.  331 

Finally,  in  the  same  conditions,  1  gramme  of  hydrogen  occupies 
a  volume  I.    The  ratios 


are  integers  which  simplify  greatly  the  chemical  equations;  if,  for 
example,  the  gas  al  obeys  the  laws  of  Avogadro  and  of  Ampere, 
the  mass  wl  of  this  gas  occupies,  in  the  normal  conditions  of  tem- 
perature and  pressure,  the  same  volume  as  2  grammes  of  hydrogen; 
hence  we  have 

ajjO^  22,    <r    "^  =  271! 

More  generally,  Vlt  V2,  .  .  .  F/,  72',  ...  are  the  numbers 
written  by  chemists  wh  n  they  wish  to  express  by  volume  the  re- 
action which  the  chemical  equation  considered  expresses  by  weight. 

250.  Law  of  equilibrium  for  the  systems  studied.  —  Suppose 
the  system  in  equilibrium  at  the  absolute  temperature  T  in  a 
vessel  where  it  is  either  alone  or  in  the  presence  of  perfect  gases 
which  do  not  take  part  in  the  reaction  ;  let  plf  p2,  .  .  .  p/,  p/  .  .  .  , 
be  the  partial  pressures  of  the  gases  al;  a2,  .  .  .  o/o/,  ...  in  this 
mixture;  these  pressures  satisfy  the  EQUILIBRIUM  CONDITION 

(3) 


M,  N,  and  Z  being  three  constants.  The  symbols  log  represent  the 
common  logarithms. 

The  pressures  plt  p2,  .  .  .  p/,  p/,  ...  are  expressed  in  terms 
of  any  unit  desired;  the  choice  of  this  unit  influences  only  the 
value  of  the  constant  Z. 

251.  Heats  of  reaction  at  constant  pressure  and  at  constant 
volume.  —  Imagine  now  that  in  a  system  sensibly  in  equilibrium  a 
small  mass  //  passes,  without  change  of  temperature,  from  the 
state  represented  by  the  first  member  of  the  chemical  equation 
to  the  state  represented  by  the  second  member  of  the  same  equa- 


332  THERMODYNAMICS  AND  CHEMISTRY. 

tion;  if  the  modification  takes  place  at  constant  pressure,  it  liber- 
ates a  quantity  of  heat  I///;  if  it  takes  place  at  c  nstant  volume, 
it  sets  free  a  quantity  of  heat  ^;  the  quantities  L  and  X,  or,  better, 
the  quantities  PL  and  PX,  which  are  the  quantities  usually  made 
use  of  in  treatises  on  thermochemistry,  are  determined  by  the  two 
following  expressions  : 


(4)  PL= 

(5)  PA-jt#*Ft'+F,'+.  .  .-V.-7,-.  .  .)r-0.4301Af]. 


nQ  is  the  normal  pressure  (as  atmospheric  pressure)  and  TQ  the 
normal  temperature  (for  instance,  that  of  melting  ice)  which  give 
to  the  specific  volume  of  hydrogen  the  value  I. 

^T 

The  constant  quotient  7^-^  is  in  the  C.G.S.  system  very  nearly 

1      Q& 

equal  to  1.  If,  therefore,  the  quantities  PL  and  PA  are  expressed 
in  small  calories,  we  may  replace  the  formulae  (4)  and  (5)  by  the 
formulas 


(40  PL 

.  .  .-Vr-.  .  .-0.4301M. 


252.  Tensions  of  saturated  vapor.     A.  Dupre*'s  formula.  — 

Let  us  indicate  by  some  examples  the  applications  which  may 
be  made  of  the  preceding  formulae. 

The  simplest  cas  is  that  where  a  single  gaseous  substance 
figures  in  the  system;  the  type  of  a  transformation  of  this  kind 
is  furnished  by  the  condensation  of  water  vapor,  represented  by 
the  chemical  equation 

H20  (vapor)  =H20  (liquid). 

Water  vapor  obeying  Avogadro's  law,  and  a  single  molecule 
of  water  vapor  figuring  in  the  first  member  of  the  chemical  equa- 
tion, we  shall  have  V=2  and  the  condition  of  equilibrium  (3) 
will  become 


- 


CHEMICAL  MECHANICS  OF  PERFECT  GASES.  333 

or  in  putting 

M  N  Z 

m 

(6) 


Athanase  Dupre1  was  the  first  to  propose  representing  by 
such  a  formula  the  relation  which  exists  between  the  tension  of 
saturated  vapor  of  water  or  of  any  other  liquid  and  the  tempera- 
ture. 

Can  this  formula  (6)  represent  with  sufficient  exactness  the 
tensions  of  saturated  vapors  measured  by  observers?  This  is  a 
question  treated  by  many  authors  and  which  has  been  recently 
examined  hi  a  very  thorough  manner  by  J.  Bertrand.2 

Three  observations  of  vapor  tensions,  at  different  but  known 
temperatures,  allow  determining  the  three  constants  ra,  n,  z;  it 
is  then  easy  to  calculate  the  value  of  p  that  formula  (6)  gives  corre- 
sponding to  each  value  of  the  temperature  T,  and  to  compare  the 
numbers  thus  obtained  with  the  results  of  observation. 

Let  us  take  as  example  water  vapor. 

If  the  pressure  p  is  calculated  in  millimetres  of  normal  mercury, 
the  constants  m,  n,  and  z  have  the  following  values: 

w=-2795; 
n=-       3.8682; 
z=+     17.44324. 

Bertrand  compared  in  five-degree  steps,  for  temperatures  in- 
cluded between  T=243°  (-30°  C.)  and  r=273°  (0°  C.),  then  in 
ten-degree  steps  from  T=273°  (0°  C.)  to  27=503°  (230°  C.),  the 
numbers  given  by  the  formula 

9  7Q^ 
log  p=  17.44324-  ^F-3.  8682  log  T 

with  the  results  of  Regnault's  experiments;  hi  the  following  table 
are  some  of  the  numbers  from  this  comparison: 

1  ATHANASE  DUPRE,  Theorie  m&anique  de  la  chaleur,  p.  97. 
*J.  BERTRAND,  Thermodynamique,  p.  101. 


334 


THERMODYNAMICS  AND  CHEMISTRY. 


T 

T-273. 

Observed. 

Calculated. 

243 

-  30 

0.39 

0.39 

273 

0 

4.60 

4.59 

323 

+   50 

91.98 

91.96 

373 

+  100 

760.00 

763.04 

423 

+  150 

3581.2 

3608.48 

473 

+  200 

11689.0 

11701.72 

483 

+  210 

14324.8 

14297.12 

493 

+  220 

17390.4 

17306.72 

503 

+  230 

20926.4 

20957.88 

The  maximum  error,  adds  Bertrand,  is  169  millimetres  for 
!F=5030  (230°  C.)  less  than  0.01  of  the  calculated  quantity,  and 
corresponding  to  an  error  of  0°.47  in  the  temperature. 

Bertrand  obtained  analogous  results  for  the  following  liquids; 
p  is  still  reckoned  in  millimetres  of  normal  mercury: 


Name  of  Liquid. 

m 

n 

z 

Water            

-2795 

-3  8682 

17.44324 

Ether       

-  1  729  97 

-  1  .  97  87 

13.433115 

Alcohol  

-  2  743  .  84  2 

-4.22482 

21.446868 

Clilorohydric  ether 

-1  747  13 

-3  8721 

17  04235 

Chloroform                            .    .    . 

-2  179  142 

—  3  91  584 

19.297930 

Carbon  bisulphide            

-1  684 

-1  7689 

12  .  58  852 

Carbon  chloride     

-2226  8 

-3.94567 

19.28670 

Sulphurous  acid  

-  1  604  8 

-3.21  98 

16.99036 

Ammonia          ,  

-  1  449  .  83 

-  1  .  87  26 

13.37156 

Nitrosren  protoxide 

+    328  05 

+  8  71  19 

-17  987082 

Carbonic  acid                        . 

-    819  77 

+  0  41  861 

6  41  443 

Turpentine                            .        ... 

—  2674  9 

-3  7283 

18  88373 

Sulphuretted  hydrogen  

-    992  6 

-0  51  415 

8  80739 

Methyl  alcohol  .               

-2661  25 

-4  6336 

22  43  1907 

Mercury  .            

-2010  25 

+  3  8806 

—   4.79892 

Sulphur  

-4684  492 

—  3.40483 

19.10740 

E.  Riecke  *  has  shown  that  the  tensions  of  vapor  of  liquid  white 
phosphorus  measured  by  Troost  and  Hautefeuille  could  be  repre- 
sented by  Dupre's  formula. 

253.  Dissociation  tensions. — Dupre's  formula  must  apply 
likewise,  as  is  evident,  to  dissociation  phenomena,  when  a  single 
gas  is  concerned  in  the  reaction.  Bertrand  showed,  in  fact,  that 
formulae  of  this  type  could  represent  in  a  satisfactory  manner  the 


1  E.  RIECKE,  Zeitschrift  fur  physikalische  Chemie^  v.  7,  p.  115,  1891. 


CHEMICAL  MECHANICS  OF  PERFECT  GASES. 


335 


dissociation  tensions  of  certain  ammoniacal  chlorides  studied  by 
Isambert.  Nevertheless,  Isambert's  determinations  being  not 
accurate,  this  verification  had  only  a  doubtful  value.  Joannis  and 
Croizier1  have  takan  up  the  study  of  the  dissociation  of  ammo- 
niacal silver  salts;  they  find  that  the  laws  of  this  dissociation  are 
very  exactly  expressed  by  formula  (6),  with  the  following  values 
of  the  constants  m,  n,  and  z;  the  pressures  p  are  reckoned  in  centi- 
metres of  normal  mercury : 


Salt  studied. 

m 

n 

z 

Temperature 
Limits  (C). 

AgBr,3NH3 

-   1787.1294 

+   1.075 

+     5.7148 

0°  to  +21° 

AgBr,|NH, 

-   6650.6086 

-35.239 

+  111.1904 

+  30    to  +55 

AgBr,NH, 
AgI,NH3 
AgI,iNH, 

-   4033.0512 
Samefo 
-  3438.3604 

-13.2489 
rmula  as  for  Ag 
-  8.8803 

+   47.5847 
Br,NH3 
+  34.0799 

+  18    to  +  64 
+  26    to  +100 

Ag€N,NH, 
AgNO,,3NH3 

-12497.1255      -58.7176 
-  5864.6826      -26.1384 

+  186.3546 
+   85.3665 

+  81    to  +117 
+  15    to  +83 

Joannis  2  has  also  studied  with  much  care  the  dissociation  of 
sodaminonium  and  potassammonium. 

He  found  that  the  dissociation  tensions  of  scdammonium, 
expressed  in  centimetres  of  mercury,  were  very  exactly  represented 
between  the  temperatures  of  -783  C.  and  +26°.21  C.  by  the 
formula 


logp=- 


619.9625 


+  5.055364  log  T-  7.814314. 


Between  the  temperatures  of  -20°  C.  and  +35°.  15  C.  the 
dissociation  tensions  of  potassammonium,  measured  in  the  same 
unit,  are  very  exactly  represented  by 


log  p= 


+  11.775  log  77-27.7003. 


254.  Guldberg  and  Waage's  law.— Let  us  return  to  the  general 
case  where  the  system  encloses  any  number  whatever  of  gases 
taking  part  in  the  reaction. 

1  JOANNIS  and  CROIZIER,  Memoires  de  la  Societt  des  Sciences  physiques  et 
naturelles  de  Bordeaux,  4th  S.,  v.  5,  p.  41,  1895. 
8  JOANNIS,  ibid.,  4th  S.,  v.  5,  p.  218,  1895. 


336  THERMODYNAMICS  AND  CHEMISTRY. 

If  the  temperature  rests  constant,  the  second  member  of  equa- 
tion (3)  remains  constant;  the  same  is  therefore  true  of  the  first 
member;  now  the  first  member  of  equation  (3)  may  in  virtue  of 
the  elementary  properties  of  logarithms  be  regarded  as  the  loga- 
lithm  of  the  following  number: 


This  number  remaining  constant  at  the  same  time  as  its  loga 
rithm,  we  have  the  following  proposition: 

//  various  systems,  in  which  a  reaction  may  be  produced  which, 
for  all  these  systems,  is  represented  by  the  same  equation,  are  in 
equilibrium  at  the  same  temperature,  the  ratio 


has  the  same  value  for  all  these  systems. 

This  law  is  known  under  the  name  of  NAUMANN'S  LAW  or 
GULDBERG  AND  WAAGE'S  LAW;  it  is  actually  a  special  case  of  the 
law  discovered  by  the  two  last  named,  and  was  not  stated  by  the 
first. 

255.  Various  examples:  Ammonium  carbonate. — Chemists 
have  made  several  interesting  verifications  of  this  law;  thus  H. 
Pelabon  has  applied  this  relation  to  certain  systems  in  which  four 
gases  enter;  such  is  the  system  composed  of  realgar,  sulphuretted 
hydrogen,  hydrogen,  and  arsenic; 1  or  again  the  system  formed 
by  mercury  sulphide,  hydrogen,  sulphuretted  hydrogen,  and 
mercury.2  We  shall  limit  ourselves  here  to  more  simple  cases. 

The  first  phenomenon  to  which  we  shall  apply  this  law  is  the 
formation  of  solid  ammonium  carbonate  at  the  expense  of  ammonia 
gas  (1)  and  carbonic  acid  gas  (2) 

The  following  is  the  reaction  equation: 

2NH3 + CO, = NH4OCONH2. 

1  H.  PELABON,  Comptes  Rendus,  v.  132,  p.  774,  1901. 
9  Ibid.,  v.  132,  p.  1411,  1901. 


CHEMICAL  MECHANICS  OF  PERFECT  GASES.          337 

Two  molecules  of  ammonia  (nj=2)  and  one  molecule  of  car- 
bonic anhydride  (n2=  1)  take  part  in  it;  the  ammonia  and  carbonic 
anhydride  obeying  Avogadro's  law,  we  have 


If,  therefore,  several  systems  enclosing  solid  ammonia  carbonate, 
ammonia  gas  whose  partial  pressure  is  plt  and  carbonic  acid  gas 
of  partial  pressure  p2  are  in  equilibrium  at  the  same  temperature, 
the  product  p^p2z=(P\P^  will  have  the  same  value  for  all  these 
systems;  it  amounts  to  the  same  thing  as  saying  that  the  product 

(8)  A"ft 

has  the  same  value  for  all  these  systems. 

Suppose  to  start  with  that  the  carbonate  dissociates  in  an 
enclosure  empty  at  first;  the  system  is  divided  into  two  phases, 
the  solid  carbonate  and  the  gaseous  mixture;  it  contains,  further- 
more, but  a  single  independent  component,  for  each  molecule  of 
carbonic  anhydride  which  it  contains  is  accompanied  by  two 
molecules  of  ammonia  ;  if  the  mass  of  the  first  substance  it  contains 
is  known,  in  the  free  state  or  in  combination,  the  mass  of  second 
substance  contained  in  it  is  also  known;  the  system  is  therefore 
mono  variant;  at  each  temperature  T  equilibrium  is  maintained 
by  a  well-determined  pressure  n. 

This  pressure  n  is  the  sum  of  the  two  partial  pressures  plt  p2; 
also,  since  the  mixed  gas  encloses,  in  this  case,  two  molecules  of 
ammonia  gas  for  one  molecule  of  carbonic  acid  gas,  the  pressure  pl 
is  twice  the  pressure  p2,  so  that 


and  the  value  of  the  product  p*p2  is,  in  this  case   equal  to  •=; 

Zif 

this  will  be  also  its  value  under  all  circumstances  at  the  same  tem- 
perature. 

Suppose  the  system  studied  encloses  not  only  two  molecules 
of  ammonia  per  molecule  of  carbonic  anhydride,  but  also  an  excess 
of  ammonia  gas;  at  the  temperature  of  the  experiment  this  extra 
gas,  distributed  throughout  the  volume  given  over  to  the  gaseous 


338  THERMODYNAMICS  AND  CHEMISTRY. 

mixture,  would  exercise  a  pressure  there  of  value  o^;  if,  in  the 
system  in  equilibrium  at  the  temperature  considered,  the  carbonic 
acid  gas  has  a  partial  pressure  p2,  the  ammonia  gas  exercises  a 
partial  pressure  Pi=2pi+w1;  as  the  total  pressure  <f>  of  the  gaseous 
mixture  is  equal  to  the  sum  (Pt+Pz),  it  is  evident  that  we  have 


The  product  p^  has  the  value  (29  +  ^i)  (9-^1).    we  also 


47T3 

know  that  it  must  have  the  value  -^=-  \  hence  the  following  equality 

£t 

is  obtained: 

(9)  (2<£+*,1)2(<5&-u1)  =  47r8. 

Suppose  now  that  for  two  molecules  of  ammonia  the  system 
encloses  not  only  one  molecule  of  carbonic  anhydride,  but  an  excess 
of  carbonic  acid  gas  besides;  the  excess  of  carbonic  anhydride 
distributed,  at  the  temperature  of  the  exper'ment,  in  the  volume 
occupied  by  the  gaseous  mixture,  will  exert  a  pressure  a>2  there; 
if,  with  the  system  in  equilibrium,  the  ammonia  gas  exerts  a  par- 
tial pressure  pl}  the  carbonic  acid  gas  exerts  a  partial  pressure 


7?2=§  +  w2;  and  as  the  sum  (pi  +  p2)  must  always  be  equal  to  the 

2 

total  pressure  <j>  of  the  gaseous  mixture,  we  have 


The  product  p^2p2  has  the  value  JJL  ^vW  -  ^L}  an(j  js 

4^3 
equal  to  -^;  therefore  it  follows  that 

(10)  (<t>-a>2) 


The  values  of  the  quantities  cjlt  cu2,  n,  being  readily  accessible 
to  obervation,  equations  (9)  and  (10)  may  be  verified. 


CHEMICAL  MECHANICS  OF  PERFECT  GASES. 


339 


Naumann  and  Horstmarm  were  the  first  to  attempt  this  verifi- 
cation; more  exact  experiments  have  been  performed  by  Isambert.1 

Five  barometer  tubes,  graduated  in  tenths  of  a  cubic  centi- 
metre, and  containing  ammonia  carbonate,  were  placed  side  by 
side  in  a  heating-chamber;  the  first,  containing  no  excess  of  car- 
bonic acid  gas  nor  of  ammonia,  gave  directly  the  dissociation  ten- 
sion it  of  ammonia  carbonate  in  vacuo  at  the  temperature  of  the 
oven;  these  results  are  found  in  column  I  of  the  following  table. 

The  four  other  tubes  contain  either  an  excess  of  carbonic 
anhydride  or  of  ammonia. 

Tube  II  had  received  an  excess  of  carbonic  acid  gas  occupying, 
in  the  normal  conditions  of  temperature  and  pressure,  16.9  cc.; 
tube  III  had  received  similarly  6.1  cc.  of  carbonic  anhydride; 
tube  IV,  6  cc.  of  ammonia  gas;  finally,  tube  V,  11.4  cc.  of  the  same 
gas;  the  readings  of  tubes  II  and  III,  joined  to  equation  (10), 
give,  at  each  temperature,  two  indirect  determinations  of  TT,  put 
in  columns  II  and  III  of  the  table;  the  readings  of  tubes  IV  and  V, 
taken  with  equation  (9),  give  for  each  temperature  two  other 
indirect  determinations  of  TT,  inscribed  in  columns  IV  and  V  of  the 
table. 

The  values  of  n  determined  indirectly  by  the  last  four  tubes 
differ  in  general  but  slightly  from  the  value  of  r  observed  directly 
by  means  of  the  first.  We  may  therefore  regard  Isambert's  ob- 
servations as  confirming  very  exactly  the  law  as  stated. 


Temperatures 
Centigrade. 

I. 

mm. 

II. 

mm. 

III. 

mm. 

IV. 
nun. 

V. 

mm. 

34°.  0 

169.8 

170.4 

164.5 

166.8 

181.3 

37  .2 

211.0 

210.8 

204.6 

205.9 

215.5 

39  .1 

234.1 

234.4 

228.5 

229.4 

236.9 

41  .8 

269.4 

271.7 

267.7 

265.6 

274.5 

42  .5 

288.3 

289.2 

284.2 

286.2 

291.9 

43  .9 

313.8 

314.5 

311.8 

313.5 

318.4 

46  .9 

375.7 

375.3 

372.0 

375.6 

378.3 

50  .1 

453.8 

452.9 

452.2 

454.1 

455.0 

52  .6 

526.2 

523.7 

522.3 

523.8 

526.2 

256.  Ammonium  cyanide. — Isambert  has  likewise  studied  the 
dissociation  of  certain  solid  non-volatile  substances,  formed  by 


1  ISAMBERT,  Comptes  Rendus,  v.  93,  p.  731,  1881 ;  v.  97,  p.  1212,  1883. 


340  THERMODYNAMICS  AND  CHEMISTRY. 

the  union  molecule  by  molecule  of  two  component  gases  ;  the  follow- 
ing equations  represent  such  reactions: 

NH3+H2S  =  HNH4S; 
PhH3+HBr=PhH4Br; 
NH4CN. 


In  this  case  ^=1,  n2=l,  and  as  the  gases  studied  obey  Avo- 
gadro's  Law,  7,  =  2,  V2=2;  at  a  given  temperature  the  product 
Pi2P22  :==  (PiPz)2  nas  a  definite  value,  as  has  also  the  product  p^. 

If,  for  instance,  the  solid  considered  dissociates  in  vacuo,  its 
dissociation  tension  attains  a  value  TT,  well  determined  for  each 
temperature;  it  is  clear,  besides,  that  in  this  case  the  two  partial 

pressures  pv  p2  are  equal  to  each  other  and  equal  to  -,  so  that  the 

product  pjp2  is  equal  to  -j  .    We  may  therefore  state  the  following 

proposition  : 

Suppose  that,  at  a  given  temperature,  one  of  the  solids  we  have 
cited  exists  in  equilibrium  with  a  gaseous  atmosphere  where  the  gaseous 
components  have  the  partial  pressures  p1}  p2;  we  shall  have  the  rela- 
tion 

11)  PlP2=^> 

TT  being  the  dissociation  tension  of  the  solid,  at  the  temperature  con- 
sidered, in  an  enclosure  containing  no  gas  at  the  start. 

Isambert  has  verified  this  relation  in  studying  the  dissociation 
of  ammonia  bisulphydrate  l  and  of  the  bromhydrate  of  phos- 
phoretted  hydrogen  ;  2  he  has  made  an  especially  careful  verifica- 
tion for  the  case  of  ammonium  cyanide  3  in  the  presence  of  an 
excess  of  ammonia  gas. 

Let  MI  be  the  pressure  exerted  by  the  ammonia  gas  in  excess, 
at  the  temperature  of  the  experiment,  if  it  occupied  alone  the 

1  ISAMBERT,  Comptes  Rendus,  v.  93,  p.  731,  1881  ;  v.  94,  p.  958,  1882;  v. 
95,  P-  1355,  1882. 

Ubid.,  v.  96,  p.  643,  1883. 

3  Ibid.,  v.  94,  p.  958,  1882;  Annales  de  Chimie  et  de  Physique,  5th  S., 
v.  28,  p.  382,  1883. 


CHEMICAL  MECHANICS  OF  PERFECT  GASES. 


341 


volume  given  up  to  the  gaseous  mixture;  in  this  mixture  the 
cyanhydric  acid  has  a  partial  pressure  p2,  and  the  ammonia  gas  a 
partial  pressure  pt  which  is  evidently  equal  to  (p2-f  c^);  *ne  total 
pressure  (f>  of  the  gaseous  mixture  being  equal  to  the  sum 
we  have 


(12) 


Further,  pl  being  equal  to 


>  equation  (11)  gives 


The  measurement  of  <j>,  joined  to  the  knowledge  of  o)lt  allows 
deriving  from  equation  (12)  a  first  value  of  p2;  again,  the  measure- 
ment of  the  dissociation  tension  n  of  ammonium  cyanide  hi  an 
enclosure  empty  at  first  allows  finding  from  equation  (13)  another 
vale  of  p2,  which  we  shall  denote  by  p/;  if  the  law  we  are  con- 
cerned with  is  true,  the  two  pressures  p2,  p2'  should  be  equal  to 
each  other. 

Here  are  the  values  of  p2  and  p2'  obtained  by  Isambert: 


Temperatures 
Centigrade. 

1C 

mm. 

* 

mm. 

<tfi 

mm. 

P2 

mm. 

Pi 

mm. 

7°.  3 

175.0 

358.0 

314.2 

21.2 

22.7 

7  .4 

176.7 

365.2 

327.7 

18.7 

21.3 

9  .2 

196.0 

369.8 

317.0 

26.4 

27.8 

9  .3 

200.0 

370.0 

329.0 

25.0 

28.0 

9  .4 

202.0 

373.4 

323.2 

25.1 

26.9 

10  .2 

214.0 

378.4 

316.0 

31.2 

32.8 

11  .0 

227.4 

393.3 

323.0 

35.1 

35.8 

11  .2 

232.9 

390.0 

311.2 

39.4 

38.7 

11  .2 

234.0 

395.6 

320.6 

37.5 

38.2 

12  .0 

246.2 

397.8 

309.2 

44.3 

42.9 

14  .3 

265.5 

413.2 

308.8 

52.2 

49.1 

15  .5 

296.9 

425.8 

294.8 

65.4 

61.8 

15  .7 

300.9 

426.1 

295.1 

65.5 

63.2 

15  .7 

300.5 

432.2 

299.8 

66.2 

62.6 

17  .0 

322.4 

441.1 

287.3 

76.9 

72.2 

17  .2 

326.2 

442.9 

286.0 

78.4 

74.0 

These  figures  place  beyond  doubt  the  exactitude  of  Guldberg 
and  Waage's  Law. 

Isambert  has  also  studied  the  dissociation  of  ammonium  cyanide 


342  THERMODYNAMICS  AND  CHEMISTRY. 

for  the  case  where  the  cyanhydric  acid  is  in  excess,  but  there  this 
substance  condenses  in  part  to  the  lipuid  state;  the  liquid  formed 
dissolves  ammonium  cyanide  and  ammonia,  and  the  conditions 
proposed  at  the  beginning  of  the  present  chapter  are  no  longer 

verified. 

257.  Influence  of  temperature.  Dissociation  of  mercuric 
oxide.—  The  various  observations  we  have  mentioned  in  the  pre- 
ceding article  show  us  that  formula  (3)  represents  very  exactly  the 
law  according  to  which  the  composition  of  the  gaseous  mixture 
within  a  system  in  equilibrium  varies  when,  without  changing  the 
temperature,  there  is  introduced  into  the  system  an  excess  of  one 
or  of  the  other  of  the  gases  which  take  part  in  the  reaction;  it 
remains  to  be  seen  if  this  formula  represents  as  exactly  the  influence 
the  temperature  exerts  on  the  degree  of  dissociation;  this  question 
has  as  yet  received  reply  only  in  the  case  (Art.  253)  where  the 
system  contains  a  single  gaseous  substance;  it  is  important  to 
examine  it  for  more  complicated  cases. 

Here  is  an  elegant  test  to  which  this  law  has  been  submitted 
by  Pelabon:1 

The  red  oxide  of  mercury  may  dissociate  into  mercury  vapor 
and  oxygen  according  to  the  equation 

HgO  =  Hg+O 

Mercury  vapor  being  monatomic  and  oxygen  diatomic,  if  we 
give  the  index  1  to  mercury  and  index  2  to  oxygen  we  shall  have 


and  equation  (3)  may  be  written 

M 
(14)  2  log  Pl+log  p2^~  +N  log  T+Z. 

Suppose,  in  the  first  place,  that  an  excess  of  mercury  is  main- 
tained in  the  system;  the  partial  pressure  pt  of  the  mercury  vapor 
in  the  gaseous  mixture  will  be  equal,  by  the  law  of  the  mixture 
of  gases  and  vapors  whose  truth  is  one  of  our  fundamental  hypoth- 

1  H.  P^LABON,  Comptes  Rendus,  v.  128,  p.  825,  1899;  Memoires  de  la 
Societe  des  Sciences  physiques  et  naturelles  de  Bordeaux,  5th  S.,  v.  5,  p.  59,  1899. 


CHEMICAL  MECHANICS  OF  PERFECT  GASES.  343 

eses,  to  the  tension  of  the  saturated  vapor  F  of  mercury  at  the 
temperature  of  the  experiment;  further,  F  is  given  by  Dupre's 
formula, 

(15)  logF=p-+nlogr+z. 

If  the  pressures  are  measured  in  millimetres  of  normal  mercury, 
the  constants  m,  nt  z  have  the  values,  as  we  have  seen,  Art.  252, 

(16)  m=2  010.25,      w=3.8806;      z=  -4.79  892. 

If,  in  formula  (14),  we  replace  log  px  by  the  value  of  log  F  given 
by  equation  (15),  and  if  we  put 

(17)  fjL=M-2m,      v=N-2n,      £=Z-2z, 
we  may  write 

(18) 


Pelabon  has  shown  that  the  values  of  p2,  expressed  in  milli- 
metres of  mercury,  could  be  very  exactly  expressed  by  a  formula 
of  this  type,  on  condition  of  taking 

(19)  /t=-27569,      v=  -57.58,      £=  +203.94711. 

Suppose  now  that  mercury  oxide  dissociates  in  an  enclosure 
which  is  at  first  a  vacuum,  and  that  the  mercury  resulting  from 
this  decomposition  remains  entirely  in  a  state  of  vapor;  it  will 
follow  necessarily,  denoting  by  p2'  the  partial  pressure  of  oxygen 
in  this  case,  that 

fc-2ft', 
and 

2  log  ft=2  log  p/+2  log  2=2  log  p2'+log  22 

=21ogp2'+log4. 

Substituting  this  value  of  2  log  pl  in  equation  (14),  we  find 


.20)  fog  ?,'=£+,/  log 


344  THERMODYNAMICS  AND  CHEMISTRY-. 

if  we  put 


or,  from  equations  (17),  if 


Equations  (16),  (19),  and  (21)  permit  us  to  calculate  the  values 
of  //,  i/,  £';  we  thus  find 

(22)         //=  -10529.8,       /=  -16.61,      f  =+  64.58  240. 

When,  therefore,  mercuric  oxide  dissociates  in  an  enclosure  at 
first  a  vacuum,  the  partial  pressure  of  the  oxygen  must,  if  the 
preceding  theory  is  correct,  be  represented  by  formula  (20),  the 
constants  ^',  i/,  £  having  the  values  given  in  (22). 

Pelabon  has  determined  experimentally  a  certain  number  of 
values  of  p2'  and  has  compared  them  with  the  values  calculated 
as  we  have  just  indicated;  the  following  table  gives  us  an  idea  of 
the  very  satisfactory  concordance  for  the  results  of  the  two  sets 
of  determinations: 


Temperatures. 

Pi  obs. 
mm. 

Pz  calc. 
mm. 

500°  C. 

995 

972 

520 

1392 

1403 

580 

3610 

3589 

610 

5162 

5308 

258.  Dissociation  of  selenhydric  acid. — Another  verification 
of  formula  (3),  verification  of  not  less  importance  than  the  pre- 
ceding, was  also  obtained  by  Pelabon  *  in  studying  the  formation 
of  selenhydric  acid  from  liquid  selenium  and  hydrogen,  according 
to  the  formula 

H2+Se=#2Se. 

1  H.  PELABON,  Comptes  Rendus,  v.  121,  p.  401,  1895;  Memoires  df  la 
Soci&e  des  Sciences  physiques  et  naturettes  de  Bordeaux,  5th  S.,  v.  3,  p.  207, 
1898. 


CHEMICAL  MECHANICS  OF  PERFECT  GASES.  345 

To  simplify  matters,  let  us  neglect  the  volatility  of  selenium; 
we  shall  then  have  in  the  system  but  two  gaseous  bodies,  hydro- 
gen, whose  partial  pressure  is  p,  and  selenhydric  acid,  of  partial 
pressure  p';  then 


2  log  p-2  log  p'=+N  loS  T+Z> 


and  equation  (3)  becomes 

or,  in  putting 

M  N 

(23)  m=~2~»      n==2' 

we  have 

(24)  log£-=^+nlo 

Pi         l 

Pelabon  found  that  the  vaues  o    —  determined  experimen- 

Pi 

tally  between  the  temperatures  320°  C.  and  720°  C  were  very 
well  represented  by  a  formula  of  the  typ  (24)  on  the  condition  of 
giving  to  the  constant  m,  n,  and  z  the  following  values; 

f  m=  13  170.3X0  4301; 

(25)  ]  n=  15.53; 

(   z=  -119  88X0.43 01. 

But  Pelabon  has  gone  farther  not  content  with  having  verified 
equation  (3),  he  has  sought  to  verify  equations  (4')  and  (S7),  which 
for  the  present  case  become  identical  with  each  other.  By  means 
of  the  equations  (23)  and  (25)  he  was  able  to  calculate: 

1°.  The  tempe  ature  for  which  L  becomes  equal  to  0,  tempera- 
lure  which  he  found  equal  to  575°  C.; 

2°.  The  heat  PL  absorbed  by  the  formation  at  15°  C.  of  a  mole- 
cule (81  grammes)  of  selenhydric  acid  at  the  expense  of  liquid 
selenium  and  hydrogen,  quantity  of  heat  which  he  found  equal 
to  17300  calories. 

The  temperature  at  which  L  becomes  equal  to  0  must,  accord- 
ing to  the  law  of  the  displacement  of  equilibrium  by  change  in 


346  THERMODYNAMICS  AND  CHEMISTRY. 

temperature  correspond  to  a  minimum  of  dissociation  or  to  a 
minimum  of  the  ratio  —f;  Pelabon  has  found,  from  direct  experi- 
ment, that  such  a  minimum  is  produced  at  a  temperature  included 
between  550°  C.  and  600"  C. 

Further,  a  direct  calorimetric  determination  gave  Fabre  *  18000 
calories  for  the  value  of  the  product  PL  at  15°  C. 

Another  verification,  analogous  to  the  preceding,  has  been 
obtained  by  Jouniaux  by  studying  the  action  of  hydrogen  on  silver 
chloride  and  the  inverse  action  of  hydrochloric  acid  on  silver.2 
The  study  of  states  of  equilibrium  which  are  established  at  tem- 
peratures included  between  525°  and  700°  allowed  him  to  deter- 
mine the  coefficients  M  N,  Z  of  the  formula  (3  He  was  then 
able  by  formula  (44)  to  calculate  the  heat  which  is  absorbed  when 
hydrochloric  acid  transforms  a  molecule  of  silver  into  a  molecule 
of  silver  chloride.  He  has  found,  for  the  value  of  this  quantity 
of  heat,  6790  calories,  while  the  direct  calorimetric  determinations 
due  to  Berthelot  gave  7000  calories. 

A  similar  investigation  concerning  the  action  of  hydrogen  on 
silver  bromide  and  the  inverse  action  gave  Jouniaux3  13700  ca- 
lories for  the  heat  of  formation  of  silver  bromide  at  the  expense 
of  hydrobromic  acid  and  silver,  while  Berthelot's  measurements 
give  14800  calories 

259>  Variations  of  the  density  of  perchloride  of  phosphorus 
vapor. — We  have  not  applied  as  yet  formula  (3)  to  equilibrium 
phenomena  which  may  be  produced  in  homogeneous  gaseous 
systems. 

Some  work  has  been  done  on  the  dissociation  of  a  gaseous 
body  into  its  gaseous  components  by  cooling  briskly  the  strongly 
heated  vessel  enclosing  the  mixture  and  analyzing  the  cooled 
mixture,  whose  composition  is  supposed  identical  with  that  of  the 
mixture  not  yet  cooled;  prolonged  experiments  by  numerous 
investigators  on  the  dissociation  of  hydriodic  acid  have  been  made 

1  FABRE,  Annales  de  Chimie  et  de  Physique,  6th  S.,  v.  10,  p.  482,  1887. 

7  A.  JOUNIAUX,  Comptes  Rendm,  v.   132,  p.   1270,   1901;     Actions    des 
hijdracides  hydrogenes  sur  I'argent  et  reactions  inverse,  p.  59    (Thesis,  Lille, 
1901). 

8  A.  JOUNIAUX,  Actions  des  hydracides  .  .  .,  p.  96. 


CHEMICAL  MECHANICS  OF  PERFECT  GASES.          347 

by  this  method;  unfortunately,  the  decomposition  of  glass,  at 
high  temperatures,  by  the  reacting  bodies  has  rendered  this  ex- 
periment untrustworthy. 

The  method  of  sudden  cooling  seems  to  be  the  only  one  which 
may  be  applied  to  the  study  of  the  dissociation  of  a  gas  which  is 
formed  without  condensation  from  its  gaseous  elements;  but  in  the 
case  where  the  gas  which  dissociates  is  formed  from  its  gaseous 
elements  with  condensation,  every  decomposition  of  this  gas  has 
for  effect  the  decrease  of  the  density  with  respect  to  air  of  the 
gaseous  mixture  where  it  exists  in  presence  of  gases  coming  from 
its  decomposition;  the  study  of  the  variations  undergone  by  the 
density  referred  to  air  of  this  mixture  when  the  temperature  and 
pressure  are  varied,  when  an  excess  of  one  or  the  other  component 
is  introduced  into  the  system,  furnishes,  on  the  subject  of  the 
dissociation  of  the  compound  considered,  indirect  but  exact  in- 
formation 

Phosphorus  perchloride  is  formed  by  the  union  of  chlorine  and 
protochloride,  according  to  the  formula 

PhCl3+Cl2=PhCl5, 

and  the  reaction  is  accompanied  by  a  condensation  which  at  con- 
stant temperature  and  pressure  reduces  by  half  the  volume  of 
the  system. 

When  one  determines,  as  did  Cahours,1  the  density  of  phos- 
phorus perchloride  vapor  at  higher  and  higher  temperatures,  this 
density  is  seen  to  diminish  more  and  more;  at  the  same  tune,  the 
vapor  assumes  a  darker  and  darker  coloration,  resembling  that  of 
chlorine ;  also,  as  soon  as  H.  Sainte  Claire  Deville  had  made  known 
the  phenomena  of  dissociation,  chemists  agreed,  with  Cannizaro 
and  H.  Kopp,  to  admit  that  the  temperature  in  increasing  brings 
about  a  gradual  dissociation  of  phosphorus  perchloride  into  chlo- 
rine and  trichloride;  this  opinion  has  been  confirmed  beyond 
dispute  by  Wanklyn  and  Robinson;8  these  observers  have  found 
that  in  diffusing  through  a  porous  substance,  the  vapors  emitted 

1  CAHOURS,  Comptes  Rendus,  v.  21,  p.  625,  1845;  Annales  de  Chimie  et 
de  Physique,  3d  S.,  v.  20,  p.  369,  1847. 

*  WANKLYN  and  ROBINSON,  Philosophical  Magazine,  v.  26,  p.  545,  1863. 


348  THERMODYNAMICS  AND  CHEMISTRY. 

by   phosphrous    perchloride   furnish   a'  mixture   which    contains 
chlorine  in  excess. 

If,  in  the  mixture  where  the  perchloride  of  phosphorus  exists 
in  the  presence  of  its  elements,  pl  denotes  the  partial  pressure  of 
trichloride,  p2  the  partial  pressure  of  chlorine,  and  p/  the  partial 
pressure  of  perchloride,  and  since  we  have 


equation  (3)  becomes 

2  log  Pl+2  log  p2-2  log  fr'^+N  log  T+Z; 

or,  in  putting 

M  N  Z 


(26)  log  Pl+log  p2-log  p/  = 

Furthermore,  an  elementary  computation  shows  that  if  we 
denote  by  dlf  d2,  £/  the  densities  of  the  three  gases  referred  to  air, 
the  density  J  of  the  mixture,  referred  to  air,  has  the  value 


ft+A+ft' 

Cahours  has  experimentally  determined  the  value  of  J  at  pres- 
sures near  to  atmospheric  pressure  and  for  temperatures  between 
182°  C.  and  336°  C.;  Troost  and  Hautefeuille  l  on  the  one  hand 
and  Wiirtz  2  on  the  other  have  made  analogous  determinations 
at  pressures  below  an  atmosphere;  finally  Wiirtz  determined  the 
vapor  density  of  a  mixture  which,  instead  of  enclosing  a  molecule 
of  phosphorus  trichloride  for  one  molecule  of  chlorine,  contained 
an  excess  of  trichloride  by  joining  to  all  these  experimental  deter- 
minations an  old  observation  of  Mitscherlich,  forty-three  values 
of  the  density  J  are  obtained  under  most  diverse  conditions;  all 
these  values,  save  one,  are,  as  shown  by  Gibbs,3  very  exactly  repre- 

1  TROOST  and  HAUTEFEUILLE,  Comptes  Rendus,  v.  83,  p.  977,  1876. 

2  WURTZ,  Comptes  Rendus,  v.  76,  p.  601,  1873. 

3  J.  WILLARD  GIBBS,  American  Journal  of  Arts  and  Sciences,  v.  18,  p.  381 
1879. 


CHEMICAL  MECHANICS  OF  PERFECT  GASES.    349 

sented  by  formula  (26)  and  (27),  on  condition  of  giving  the  con- 
stants //,  v,  and  £  properly  chosen  values. 

260.  Dissociation  of  a  gas  into  a  vacuum  space. — The  study 
of  vapor  densities  of  phosphorus  perchloride  furnishes  also  a  re- 
markable confirmation  of  the  theory  of  dissociation  within  a 
system  containing  a  mixture  of  perfect  gases.  Let  us  discuss  the 
consequences  to  which  this  theory  leads  for  the  case  where  the 
compound  dissociates  in  a  vacuum  enclosure,  so  that  the  gases 
coming  from  this  decomposition  exist  in  the  system  in  just  the 
proportion  as  in  the  compound  itself. 

Let  us  denote  by  x  the  mass  of  non-dissociated  gas  contained 
in  1  gramme  of  the  gaseous  mixture;  x  will  be  equal  to  1  when  the 
combination  is  entire  and  to  0  when  the  decomposition  is  com- 
plete; suppose  that,  leaving  the  pressure  at  a  constant  value  TT, 
the  absolute  temperature  T  increases  from  0  to  +  oo ,  and  let  us 
see  how  x  varies  according  to  formula  (3). 

The  problem  is  particularly  simple  for  the  case  in  which  the 
compound  is  formed  without  condensation  from  its  elements; 
in  this  case  the  heat  of  formation  under  constant  pressure  and 
the  heat  of  formation  at  constant  volume  of  the  compound  sub- 
stance are  equal  to  each  other,  by  definition;  besides,  we  know 
(Chap.  Ill,  Art.  44)  that  they  are  both  independent  of  the  tem- 
perature, by  virtue  of  the  equality  (4) ;  we  must  have,  therefore 


There  are  then  two  cases  to  distinguish:  either  M  is  negative 
and  the  compound  is  unceasingly  exothermic,  or  else  M  is  positive 
and  the  compound  is  constantly  endothermic. 

In  the  one  case  as  in  the  other,  the  second  member  of  equation 

(3)   reduces  to    f -™-+ZJ,  which  permits  establishing  the  three 

following  laws: 

FIRST  CASE:  THE  COMPOUND  is  EXOTHERMIC. — Draw  two 
rectangular  axes  OT,  Ox  (Fig.  Ill);  lay  off  as  abscissae  the  abso- 
lute temperatures  T,  and  as  ordinates  the  values  of  z;  for  T=0T 
x  has  the  value  1 ;  the  curve  representing  the  variations  of  x  starts 
from  the  point  A,  in  very  close  contact  with  the  straight  line  A  A', 
which  has  the  constant  ordinate  x=l;  it  is  only  after  a  consider- 


350 


THERMODYNAMICS  AND  CHEMISTRY. 


able  distance  AB  that  it  separates  appreciably  from  this  straight 
line;  it  then  descends  from  left  to  right  along  BC,  and  when  T 
increases  beyond  all  limits,  it  approaches,  without  reaching,  a 
straight  line  LL'  parallel  to  OT;  this  straight  line  has  a  constant 
ordinate  greater  than  0,  so  that,  the  temperature  increasing  without 
limit,  the  system  does  not  approach  the  state  of  complete  dissociation. 

The  curve  representing  the  variations  of  x  does  not  change 
if  the  value  of  the  constant  pressure  changes  under  which  the 
experiment  is  supposed  to  be  made. 


FIG.  111. 


FIG.  112. 


SECOND  CASE:  THE  COMPOUND  is  ENDOTHERMIC. — The  ratio 
x  starts  for  ^=0  from  the  value  0;  the  curve  (Fig.  112)  representing 
the  variations  of  x  starts  from  the  point  0;  it  has  close  contact 
with  the  line  OT  so  that  it  does  not  sensibly  leave  this  curve  until 
after  the  considerable  distance  OB;  it  then  rises  from  left  to  right 
along  BC,  and.  when  the  temperature  becomes  indefinitely  great, 
it  approaches  without  reaching  a  straight  line  LL'  parallel  to  OT] 
this  line  LL'  lies  below  the  straight  line  AA',  whose  constant  ordi- 
nate is  z=l;  consequently  when  the  temperature  increases  indefi- 
nitely the  state  of  the  system  does  not  approach  a  complete  combina- 
tion. 

The  curves  representing  the  variations  of  x  do  not  change  if  the 
value  of  the  constant  pressure  changes  under  which  the  experi- 
ment is  supposed  to  be  made. 

For  the  case  in  which  the  compound  studied  is  formed  with 
condensation  the  results  become  slightly  more  complicated;  the 
path  of  the  representative  point  is  no  longer  independent  of  the 
pressure;  it  is,  on  the  contrary,  the  higher  as  the  pressure  is  greater. 

Suppose  in  particular,  following  Gibbs,  that  the  constant  N 
be  here  also  equal  to  0 ;  the  heat  of  formation  under  constant  pres- 
sure will  no  longer  depend  upon  the  temperature;  it  will  be  a 


OF  THE 


CHEMICAL  MECHANICS  OF^H&Oi^^^KBS.  351 

simple  constant;  consider  only  the  case  where,  M  being  negative, 

the  COMPOUND  IS  CONSTANTLY  EXOTHERMIC. 

When  the  temperature  T  starts  from  zero  and  increases  beyond 
limit,  the  pressure  keeping  an  in-  a? 
variable  value  TT,  x  starts  from  the 
value  1  and  the  curve  representing 
the  variations   of   x  (Fig.  113)  be- 
gins at  the  point  A,  whose  ordinate 
OA  is  equal  to  unity. 

When  the  temperature  rises,  the 
curve  remains  for  quite  a  long  dis-  FlG-  113- 

tance  AB  identical  with  the  straight  line  AA'  drawn  through  the 
point  A  parallel  to  OT;  it  then  detaches  itself  to  descend  from 
left  to  right  along  BC;  when  the  temperature  increases  without 
limit,  it  approaches  more  and  more  a  line  LU  parallel  to  OT,  but 
situated  above  OT7. 

Under  another  constant  pressure  a),  less  than  TT,  things  go  on 
in  a  similar  manner,  but — 

1°.  The  curve  detaches  itself  from  the  straight  line  AA'  at  a 
point  6,  situated  to  the  left  of  the  point  B; 

2°.  The  curve  be  is  constantly  below  the  curve  BC 

3°.  When  the  temperature  T  increases  without  limit,  the  line 
be  approaches  more  and  more  the  line  U',  parallel  to  LLr,  but 
situated  between  OT  and  LU. 

261.  Variations  of  vapor  density.  Are  they  due  to  the  dis- 
sociation of  polymers  ? — It  is  in  the  study  of  the  great  variations  of 
density  of  certain  vapors  that  the  preceding  principles  find  their 
chief  use. 

If,  in  various  conditions  of  temperature  and  pressure,  the  density 
of  a  sensibly  pe  feet  gas  referred  to  air  is  determined,  the  same 
number  is  always  found ;  this  density  is  a  constant.  This  will  not 
be  true  for  a  gas  which  is  appreciably  other  than  a  perfect  gas; 
for  example,  the  density  of  carbonic  acid  gas  referred  to  air,  at 
atmospheric  pressure,  is  somewhat  less  at  100°  than  at  0°. 

The  density  referred  to  air  of  certain  gases  or  of  certain  vapors 
undergoes  very  great  variations  when  the  temperature  and  pres- 
sure are  changed;  the  first  observation  of  such  variations  was 
made  in  1844  by  Cahours,  who  found  the  vapor  density  of  acetic 


352  THERMODYNAMICS  AND  CHEMISTRY. 

acid,  taken  at  atmospheric  pressure,  to  vary  from  3.20  to  2.08, 
while  the  temperature  rose  from  125°  C.  to  338°  C. 

Since  this  time,  analogous  facts  have  multiplied;  formic  acid 
and  nitrogen  peroxide  show  similar  variations  to  those  manifested 
by  acetic  acid;  Troost  and  Hautefeuille  have  proved  that  the 
density  of  sulphur  vapor,  at  atmospheric  pressure,  passed  sensibly 
from  the  value  6.6  to  the  value  2.2  when  the  temperature  went 
from  500°  C.  to  1000°  C.;  the  experiments  of  Crafts  and  Meier, 
performed  by  the  method  of  displacement  of  air,  showed  that 
iodine  vapor  density  nearly  constant  and  equal  to  8.8  while  the 
temperature  remained  less  than  700°  C.,  rapidly  decreased  beyond; 
reaching  a  value  slightly  greater  than  4.4  when  the  temperature 
exceeded  1600°. 

We  may  be  content  with  establishing  these  facts  and  saying 
that  the  gases  or  vapors  whose  density  referred  to  air  undergoes 
great  changes,  caused  by  variations  in  pressure  and  temperature, 
are  far  removed  from  the  state  of  perfect  gas. 

Certain  physicists  have  thought  a  more  complete  and  fruitful 
interpretation  of  these  variations  could  be  sought:  they  have 
regarded  the  gases  where  they  are  manifested  as  capable  of  existing 
in  two  different  states;  when  each  of  these  two  gaseous  states  is 
sensibly  perfect,  its  density  referred  to  air  is  sensibly  independent 
of  the  temperature  and  pressure;  but  the  densities  referred  to  air 
of  these  two  gases  are  different ;  they  are  in  a  simple  ratio  to  each 
other;  a  gas  whose  density  changes  considerably  with  the  tem- 
perature and  pressure  is  in  reality  a  mixture  of  two  gases  one  of 
which  is  a  polymer  of  the  other,  and  the  proportions  of  this  mixture 
vary  with  the  temperature  and  pressure. 

Thus,  according  to  this  hypothesis,  there  exist  two  iodine  vapors, 
one  of  which  alone  would  have  the  density  8.8  and  to  which  Avo- 
gadro's  Law  would  assign  the  symbol  72,  while  the  other  would 
have  the  density  4.4  and  the  symbol  7  by  the  same  law;  there 
would  exist  two  acetic  acid  gases,  two  formic  gases,  two  peroxides 
of  nitrogen,  the  density  of  one  of  these  two  gases  being  double 
the  density  of  the  other. 

There  are  a  great  number  of  cases  where  the  existence  of  two 
forms  of  the  same  gas,  polymers  of  each  other,  is  incontestable; 
everybody  knows,  for  instance,  that  oxygen  exists  in  the  state  of 


CHEMICAL  MECHANICS  OF  PERFECT  GASES.          353 

ordinary  oxygen  and  ozone;  according  to  Soret's  researches  the 
density  of  ozone  is  f  that  of  oxygen;  Avogadro's  Law,  which  gives 
oxygen  the  symbol  O2,  gives  ozone  the  symbol  O3.  Thus,  in  this 
case,  we  may  observe,  at  the  same  temperature  and  pressure, 
samples  of  oxygen  which  have  different  densities,  different  chemi- 
cal and  physical  properties,  so  that  there  is  no  doubt  of  the  exist- 
ence of  an  allotropic  oxygen. 

Organic  chemistry  furnishes  us  with  a  great  number  of  analo- 
gous facts;  thus  every  chemist  knows  that  acetylene  may  be 
transformed  into  a  polymer  of  triple  density,  benzine. 

But  if,  in  these  various  cases,  we  can  put  beyond  doubt  the 
existence  of  the  same  gas  in  two  distinct  polymeric  forms,  we  are 
indebted  to  the  phenomena  of  false  equilibrium;  in  the  conditions 
of  temperature  and  pressure  for  which  the  states  of  false  equilib- 
rium would  not  be  produced,  oxygen,  taken  in  definite  conditions, 
would  always  enclose  a  determined  amount  of  ozone;  at  a  given 
pressure  and  temperature  its  properties  would  be  perfectly  deter- 
mined; but  its  density  taken  with  respect  to  a  perfect  gas  would 
vary  with  the  pressure  and  with  the  temperature;  oxygen  would 
behave,  in  terms  of  the  variation  of  density  produced  by  a  rise  in 
temperature,  just  as  do  sulphur  vapor,  iodine  vapor,  acetic  acid 
vapor;  one  may  not,  therefore,  argue  from  this  fact  that  at  a  given 
pressure  and  temperature  each  of  these  gases  exists  in  a  perfectly 
determined  state  in  order  to  deny,  for  each  of  them,  the  coex- 
istence of  two  polymers;  one  may  merely  conclude  there  are  not 
produced  phenomena  of  false  equilibrium  in  the  conditions  of 
temperature  and  pressure  for  which  the  experiments  have  been 
performed. 

262.  Comparison  of  experimental  facts  with  the  theory  of 
dissociation. — To  explain  the  variations  undergone  by  the  density 
referred  to  air  of  certain  gases,  when  the  temperature  and  pressure 
change,  by  regarding  each  of  these  gases  as  a  mixture  of  two  states 
one  of  which  is  a  polymer  of  the  other,  is  to  make  an  hypothesis 
in  no  way  unacceptable ;  this  hypothesis  will  assume  a  high  degree 
of  probability  if  it  be  shown  that  in  applying  to  such  a  mixture 
the  thermodynamic  properties  of  a  mixture  of  gases  near  to  the 
perfect  state,  one  succeeds  in  taking  account  of  these  changes  of 
density  in  a  complete  manner. 


354  THERMODYNAMICS  AND  CHEMISTRY. 

Suppose  the  polymer  formed  with  liberation  of  heat  and,  to 
simplify,  admit,  as  does  Gibbs,  that  the  constant  N  equals  0;  if 
x  denotes  the  mass  of  the  polymer  contained  in  1  gramme  of  the 
gaseous  mixture,  the  variations  undergone  by  x  when  the  tem- 
perature increases,  keeping  the  pressure  constant,  will  be  represented 
by  one  of  the  curves  ABC  or  Abe  of  Fig.  113.  Observe  now  that  the 
density  J  of  our  complex  gas  referred  to  air  increases  constantly 
with  x,  starts  for  x=0  from  the  density  d  of  the  non-polymerized 
gas,  and  attains  for  x  =  1  the  density  D  of  the  polymer,  and  we  may 
state  the  following  results: 

Take  two  rectangular  axes  (Fig.  114);  on  the  axis  of  abscissae 
lay  off  the  absolute  temperatures  T,  and 
on  the  axis  of  ordinates  the  densities  A ; 
take  an  invariable  value  n  of  the  pres- 
sure, and  let  the  temperature  T  increase 
from  0  to  +  oo .  The  representative 
point  starts  from  D,  whose  ordinate  OD 
is  measured  by  the  density  D  of  the 
polymer;  the  curve  which  it  describes 
remains,  for  a  conside  able  distance  DB,  FJG  114 

almost  identical  with  the  straight  line 

DD'  drawn  through  the  point  D  parallel  to  OT;  it  then  descends 
from  left  to  right  along  BC;  when  T  increases  indefinitely,  the 
representative  point  approaches  a  line  RR'f  parallel  to  OT,  but 
whose  constant  ordinate  surpasses  the  density  d  of  the  non-poly- 
merized gas. 

If  the  same  observations  are  repeated  under  constant  pressure 
w,  less  than  it,  one  finds  repersenting  them  an  analogous  curve 
Dbc,  but— 

1°.  The  point  6,  where  the  curve  Dbc  quits  appreciably  the  line 
DD',  is  situated  to  the  left  of  the  point  B; 

2°.  The  line  be  is  constantly  below  the  line  BC 

3°.  When  T  increases  indefinitely,  the  representative  point 
approaches  a  line  rr',  parallel  to  OT,  and  situated  between  RR' 
and  dd'. 

263.  Density  of  iodine  vapor. — The  arrangement  we  have 
just  described  is  exactly  that  of  the  curves  by  which  Crafts  and 


CHEMICAL  MECHANICS  OF  PERFECT  GASES.    355 

Meier  l  have  represented  the  variations  undergone  by  iodine  vapor 
when  the  temperature  varies  from  500°  to  16CKP,  at  constant  pres- 
sure, to  which  Crafts  and  Meier  have  given  successively  the  values 
0.4  atm.,  3.0  atm.,  0.1  atm. 

264.  Gibbs's  formula.  —  But  we  are  not  satisfied  with  this 
merely  qualitative  comparison  between  the  results  of  formula  (3; 
and  the  experimental  data. 

Take,  for  instance,  the  polymerization  of  nitrogen  peroxide, 
represented  by  the  formula 

2NO2=2N204. 
We  have  here  V\=4,  TY=2,  so  that  formula  (3)  may  be  written 


Let  D  and  d  be  the  densities  referred  to  air  of  the  substances 
N2O4  and  NO2,  the  first  being  double  the  second.  The  equality 
(27)  gives  us 


Furthermore,  denoting  by  TT  the  total  pressure,  we  have 

These  two  equations  of  the  first  degree  in  pt  and  p/  give  us 
D-A      D-A 

,^J-d_^4-d^ 

Equation  (28)  then  becomes 
(29)  lo 


1  CiiAFrs  and  MEIER,  Comptes  Rendus,  v.  90,  p.  690,  1880. 


356  THERMODYNAMICS  AND  CHEMISTRY. 

Gibbs  1  took  AT=0;  that  is  to  say,  he  has  admitted  that,  at 
constant  pressure,  the  heat  of  formation  of  the  polymer  N2O4  from 
the  gas  NO2  was  independent  of  the  temperature;  he  has  shown 
that  formula  (29)  represented  in  a  satisfactory  manner  the  deter- 
minations of  vapor  density  of  nitrogen  peroxide  made  at  different 
\emperatures  and  under  different  pressures  by  Mitscherlich, 
R.  Miiller,  H  Deville  Troost,  Naumann,  Playfair,  and  Wanklyn. 

A  similar  formula  represents  the  vapor  densities  of  acetic  acid 
observed  by  Cahours,  Bineau,  Horstmann,  Troost,  Naumann, 
Playfair,  and  Wanklyn;  another  the  vapor  densities  of  formic  acid, 
determined  by  Bineau. 

After  some  very  exact  investigations  of  the  vapor  density  of 
nitrogen  peroxide  E.  and  L.  Natauson  have  observed  that 
formula  (29)  did  not  represent  with  an  entire  exactitude  the  varia- 
tions of  this  density;  but  on  the  subject  of  these  discrepancies 
there  are  two  points  to  note. 

1°.  For  simplicity,  Gibbs  has  attributed  the  value  0  to  the 
constant  N,  supposition  which  is  not  obligatory; 

2°.  The  preceding  theory  supposes  the  two  gases  N02,  N204  are 
in  the  state  of  perfect  gases,  supposition  certainly  removed  from 
the  truth. 

Gibbs's  theory  does  not  give  the  laws  of  chemical  equilibrium 
in  systems  containing  gases  more  exactly  than  the  laws  of  Boyle 
and  Charles  make  known  the  compressibility  and  expansion  of  a 
single  gas;  but  it  is  enough  that  it  renders  in  chemical  mechanics 
services  analogous  to  those  which  Boyle's  and  Charles's  laws 
render  in  physics,  to  be  extremely  valuable. 

1  GIBBS,   Transactions  of  the  Connecticut  Academy,  v.   3,  p.   234,   1876; 
American  Journal  of  Arts  and  Sciences,  v.  18,  p.  277,  1879. 

2  E.  and  L.  NATAUSON,  Wiedemann's  Annalen,  v.  24,  p.  454,  1885;  v.  27, 
p.  606,  1886. 


CHAPTER  XV  I. 
CAPILLARY  ACTIONS   AND   APPARENT   FALSE   EQUILIBRIA. 

265.  The  preceding  theories  are  often  contradicted  by  experi- 
ment.— The  comparisons  which  we  have  constantly  had  occasion 
to  make  between  the  results  of  chemical  statics  founded  on  thermo- 
dynamics and  the  experimental  data  have  revealed  numerous  and 
precise  concordances,  but  they  have  likewise  shown  evidence  of 
too  numerous  and  sharp  contradictions  for  us  to  pass  over  them 
in  silence 

The  decomposition  of  water  absorbs  heat ;  when,  therefore,  we 
raise  the  temperature  of  a  mixture  of  oxygen  hydrogen,  and  water 
vapor,  the  water  vapor  should  dissociate  more  and  more;  but  if 
we  take  a  mixture  of  oxygen  and  hydrogen  and  gradually  raise 
the  temperature,  we  shall  have  at  first  no  chemical  reaction ;  then, 
all  at  once,  when  the  temperature  attains  about  500°,  a  part  of 
the  mixture  passes  with  explosion  into  the  state  of  water  vapor. 

The  formation  of  ozone  from  oxygen  absorbs  heat;  ozone 
therefore  should  be  the  more  stable  the  higher  the  temperature; 
now  it  is  sufficient  to  heat  to  200°  ozonized  oxygen  in  order  to 
have  every  trace  of  ozone  disappear. 

All  the  explosive  reactions,  all  rapid  combustions  are  as  many 
exceptions  or,  better,  objections  to  the  principle  of  displacement  of 
equilibrium  by  variation  of  the  temperature. 

The  chemical  actions  are  not  the  only  ones  which  make  ex- 
ception to  the  rules  laid  down  by  thermodynamics;  the  physical 
changes  of  state  and  allotropic  modifications  also  furnish  objec- 
tions to  this  theory. 

According  to  this  theory,  when  a  liquid  is  transformed  into 
vapor  there  exists  for  every  temperature  one  pressure,  and  one 

357 


358  THERMODYNAMICS  AND  CHEMISTRY. 

only,  for  which  there  is  equilibrium  between  the  liquid  and  vapor; 
at  a  lower  pressure  the  liquid  should  become  vapor;  at  a  higher 
pressure  the  vapor  must  condense.  This  is  not  what  experiment 
shows  to  happen;  drops  of  water,  suspended  in  a  liquid  of  the 
same  density,  may,  without  quitting  the  liquid  state,  be  carried 
to  a  temperature  at  which  the  tension  of  saturated  vapor  exceeds 
by  a  great  deal  the  pressure  which  they  support;  very  dry  and 
quite  pure  vapor  may,  without  any  condensation  being  produced, 
be  compressed  beyond  the  tension  of  the  vapor  saturated  at  the 
temperature  of  the  experiment 

When  a  solid  and  the  liquid  resulting  from  its  fusion  are  sub- 
mitted to  atmospheric  pressure,  there  exists,  according  to  the 
preceding  theory,  a  temperature,  and  one  only,  where  the  solid  is 
in  equilibrium  with  the  liquid/  at  higher  temperatures  the  solid 
should  melt;  at  lower  temperatures  the  liquid  should  freeze. 
This  last  prediction  is  not  confirmed  by  experiment;  the  tempera- 
ture of  a  substance  may  be  lowered  far  beneath  the  freezing-point 
without  the  substance  leaving  the  liquid  state 

When  a  salt  is  in  contact  with  a  solvent  there  exists,  at  each 
temperature  and  pressure,  a  concentration  for  which  the  system  is 
in  equilibrium;  in  the  presence  of  a  less  concentrated  solution 
the  solid  salt  should  dissolve;  from  a  more  concentrated  solution 
the  dissolved  salt  should,  in  part,  be  precipitated  in  the  solid  state. 
In  this  last  point  the  theory  is  not  in  accord  with  observation; 
a  solution  may  be  kept  supersaturated  without  the  salt  contained 
in  it  crystallizing  out. 

Similarly  a  gaseous  solution  may  be  maintained  supersaturated 
in  such  conditions  of  temperature  and  pressure  that,  according  to 
thermodynamics,  the  gas  should  be  liberated 

According  to  the  phase  rule  there  should  be  a  temperature 
where  orthorhombic  sulphur  (octahedric)  and  clinorhombic  sul- 
phur (prismatic)  coexist  in  equilibrium;  above  this  temperature 
orthorhombic  sulphur  should  be  transformed  into  clinorhombic; 
below  this  temperature  the  clinorhombic  form  should  be  changed 
into  orthorhombic  sulphur.  Actually,  below  the  transformation- 
point  one  may  have  clinorhombic  sulphur  in  the  state  of  crystalline 
surfusion;  above  the  transformation  point  orthorhombic  sulphur 
may  be  had  in  the  state  of  crystalline  superheating. 


CAPILLARY  ACTIONS— APPARENT  FALSE  EQUILIBRIA.  359 

266.  Rule,   stated  by  J.  Moutier,  which  summarizes  these 
contradictions. — These  numberless  observations,  contradicting  the 
thermodynamic  theory,  all  possess  a  common  character. 

Never  do  we  meet,  in  the  domain  of  experimental  facts,  a 
modification  declared  by  thermodynamics  to  be  impossible;  we 
never  see  two  substances  combine  when  the  theory  declares  they 
will  not  combine;  a  compound  dissociate,  when  theory  affirms  it 
will  not  decompose;  a  liquid  change  into  vapor  or  freeze  when, 
according  to  thermodynamics,  it  should  not  vaporize  or  congeal; 
without  exception,  when  thermodynamics  states  a  modification  to 
be  impossible,  the  modification  is  not  produced. 

But,  on  the  other  hand,  when  thermodynamics  announces  that 
a  modification  will  be  produced,  the  modification  is  not  always 
produced. 

Thermodynamics  affirms  that  at  ordinary  temperatures  oxygen 
and  hydrogen  will  combine  almost  entirely,  that  nitre  will  decom- 
pose; but  oxygen  and  hydrogen  rest  mixed  without  combining, 
and  nitre  does  not  decompose.  Water,  by  the  retarding  of  boiling, 
should  become  vapor;  in  surfusion  it  should  freeze;  in  both  cases 
it  remains  in  the  liquid  state. 

In  conclusion,  the  following  proposition,  due  to  Moutier,  and 
already  spoken  of  in  Art.  99,  may  be  stated: 

Every  time  that  thermodynamics,  by  aid  of  the  hypotheses  and 
principles  mentioned  as  yet,  announces  that  a  certain  state  will  be, 
for  the  system  studied,  a  state  of  equilibrium,  observation  shows  that 
the  system,  put  in  this  state,  remains  there  actually  in  equilibrium; 
but  when  thermodynamics  states  that  the  system  studied,  put  in  a 
certain  state,  will  undergo  a  determined  modification,  it  may  happen 
that  the  system,  placed  in  this  state,  remains  there  in  equilibrium. 

267.  True  and  false  equilibria. — In  other  words,  experiment 
recognizes  all  the  equilibrium  states  predicted  by  thermodynamics, 
states  we  shall  call  TRUE  EQUILIBRIUM  STATES;    but  besides,  it 
recognizes  the  existence  of  a  great  number  of  equilibrium  states 
which  the  principles  of  thermodynamics  deny;   to  these  last  we 
have  given  the  name  FALSE  EQUILIBRIUM  STATES. 

268.  Internal   thermodynamic   potential   of   a  homogeneous 
mass  whose  various  particles  are  infinitely  separated. — A  theory 
permits,  in  a  great  number  of  cases,  not  only  to  comprehend  the 


360  THERMODYNAMICS  AND  CHEMISTRY. 

existence  of  false  equilibrium  states,  but  also  to  predict  the  cir- 
cumstances which  will  assure  the  maintenance  of  such  states  or 
provoke  their  rupture.  The  broad  lines  of  this  theory  were  traced 
by  J.  Willard  Gibbs  1  in  various  parts  of  his  admirable  memoir  on 
the  equilibrium  of  heterogeneous  substances. 

Take  a  certain  mass  M  of  water,  brought  to  a  definite  tempera- 
ture, say  100°,  and  having  a  certain  density.  Divide  this  mass 
into  infinitely  small  particles,  identical  with  each  other,  and  sow 
them  in  space  at  infinite  distances  from  each  other. 

This  mass  of  water,  thus  pulverized  and  disseminated,  admits 
a  certain  internal  thermodynamic  potential  £F;  one  may  evidently 
regard  this  potential  as  the  sum  of  the  internal  thermodynamic 
potentials  possessed  by  the  water  particles  if  each  of  them  existed 
alone  in  space.  Further,  as  these  small  parts  are  supposed  identical 
one  with  another,  all  these  partial  thermodynamic  potentials  must 
be  equal  to  each  other;  to  find  their  sum  it  will  be  sufficient  to 
take  the  value  for  one  of  them  and  multiply  this  value  by  the 
number  of  parts  into  which  the  mass  M  has  been  divided.  Thus 
if  g  denotes  the  internal  thermodynamic  potential  possessed  by  one 
of  our  small  masses  of  water,  in  the  conditions  indicated,  abso- 
lutely isolated  in  space,  and  if  n  denotes  the  number  of  these  small 
masses  into  which  the  mass  M  has  been  divided,  we  shall  have 


Other  things  being  equal,  the  number  n  of  parts  conforms  to 
a  given  type,  which  may  be  considered  in  the  mass  M  as  propor- 
tional to  the  magnitude  of  this  mass;  the  preceding  result  may 
therefore  be  stated  thus: 

When  the  mass  M  of  water  is  divided  into  particles  infinitely 
distant  from  each  other,  the  internal  thermodynamic  potential  of 
this  mass  is  of  the  form 


fj)  being  a  quantity  depending  solely  upon  the  temperature  and 
density  of  water. 

1  J.  WILLARD  GIBBS,  Transactions  of  the  Connecticut  Academy,  v.  3,  pp.  120 
and  416,  1876. 


CAPILLARY  ACTIONS— APPARENT  FALSE  EQUILIBRIA.  361 

269.  Internal  thermodynamic  potential  of  a  homogeneous 
mass  when  account  is  taken  of  the  arrangement  of  its  parts. — 
But  what  we  have  the  intention  to  consider  is  not  a  mass  thus 
disseminated;  it  is  a  coherent  mass  of  water,  whose  various  parts 
are  adjacent  to  each  other,  which  forms  a  continuous  whole,  limited 
by  a  certain  surface;  this  mass  evidently  cannot  be  considered  as 
being  in  the  same  state  as  the  preceding;  we  may  not  say  whether 
disseminating  the  particles  crowded  together  or  closing  in  the 
scattered  parts  is  an  operation  which  does  not  modify  our  mass 
of  water. 

Further,  if  the  mass  of  water  divided  into  infinitely  small  and 
infinitely  distant  parts  and  the  mass  of  water  for  which  these  parts 
are  collected  together  cannot  be  regarded  as  being  in  the  same  state ; 
we  cannot  affirm  without  hypothesis  that  these  two  masses  have 
the  same  internal  potential;  to  free  ourselves  from  hypotheses, 
we  must  think,  at  least  provisionally,  that  their  thermodynamic 
potentials  are  different;  that  if  M$  is  the  internal  thermodynamic 
potential  of  the  divided  mass,  the  same  mass,  brought  to  con- 
tinuity, will  have  an  internal  thermodynamic  potential  of  the  form 
(M<f>+  W)j  W  depending  not  only  on  the  density  and  temperature 
of  the  water,  but  also  on  the  arrangement  of  the  various  parts  of 
the  mass  M  or,  in  other  terms,  on  the  form  of  this  mass. 

Concerning  this  quantity  W  thermodynamics  gives  us  the 
folio  whig  information:  The  various  infinitely  small  parts  into 
which  the  mass  of  water  may  be  divided  exercise  certain  actions 
on  each  other  which  admit  a  potential,  and  W  is  precisely  this 
potential. 

We  see,  therefore,  that  the  value  of  W  will  depend  on  the  hy- 
potheses made  concerning  the  actions  exerted  upon  each  other  by 
the  infinitely  small  masses  into  which  the  total  mass  may  be 
divided. 

270.  Hypothesis  of   molecular   attraction. — Concerning    the 
subject  of  these  hypotheses  our  choice  is  determined;  since  New- 
ton, physicists  have  almost  constantly  made  two  hypotheses  con- 
cerning the  actions  which  two  material  masses  exert  on  each 
other;    the  hypothesis  of  universal  attraction  and  the  hypothesis 
of  molecular  attraction;    these  two  hypotheses  have  been  very 
fruitful  both  hi  celestial  mechanics   and  in  physical  mechanics; 


362  THERMODYNAMICS  AND  CHEMISTRY. 

it  is  then  quite  natural  to  keep  them  and  take  them  as  starting- 
point  in  the  determination  of  *F. 

These  hypotheses  may  be  formulated  thus: 

If  two  very  small  masses  m,  mf  are  separated  by  a  distance  r, 
each  of  them  exerts  on  the  other  an  attractive  force  directed  along 
the  line  joining  them. 

The  force  which  the  mass  m  exerts  on  the  mass  mf  is  equal  in 
magnitude  to  the  force  exerted  by  the  mass  m'  on  m. 

The  value  of  each  of  these  forces  is  the  sum  of  two  terms. 

The  first  term  (term  of  universal  attraction)  has  the  value 


K  being  a  positive  constant  coefficient  whose  value  depends  neither 
upon  the  nature  nor  the  condition  of  the  two  masses  m  and  mf. 
The  second  term  (term  of  molecular  attraction)  has  the  value 

mm']. 

The  value  of  the  coefficient  /  depends  not  only  on  the  distance 
r  which  separates  the  two  masses  m,  mf,  but  also  on  the  nature 
and  state  of  these  two  masses.  As  to  the  variation  undergone  by 
the  coefficient  /  when  the  distance  r  changes  in  value,  the  follow- 
ing suppositions  are  made: 

For  every  sensible  value  of  the  distance  r,  the  coefficient  /  is 
so  small  that  the  term  of  molecular  attraction  is  negligible  com- 
pared with  the  term  of  universal  attraction;  on  the  contrary, 
when  the  distance  r  becomes  inferior  to  a  certain  limit  X,  which 
is  of  an  extreme  smallness  and  which  is  called  the  radius  of  molecu- 
lar activity,  the  coefficient  /  assumes  a  very  great  value;  it  is  then 
the  term  of  universal  attraction  which  is  negligible  as  compared 
with  the  term  of  molecular  attraction. 

Such  are  the  principles  on  which  are  based  the  determination 
of  V,  thence  reduced  to  a  problem  of  mathematical  analysis. 

The  first  result  reached  by  the  mathematicians  is  the  following: 

Given  the  small  magnitude  of  masses  the  chemist  and  physicist 
have  to  deal  with,  we  may,  in  the  formation  of  W,  take  no  account 
of  the  universal  attraction  term  and  concern  ourselves  only  with 
molecular  attractions. 


CAPILLARY  ACTIONS-APPARENT  FALSE  EQUILIBRIA.  363 

This  first  point  established,   these  methods,   due  to  Gauss, 
allow  of  demonstrating  that  IF  is  o\  the  following  form: 


S  being  the  area  of  the  surface  limiting  one  mass  of  water  and  <p 
A  being  two  quantities  which  depend  upon  the  nature  of  the  water 
and  its  density. 

If  we  denote  by  ©  the  sum  (<j>+<p)  which  depends  upon  the 
temperature  and  density  of  water,  we  see  that  the  internal  potential 
of  our  mass  of  water  will  be  of  the  form  : 


271.  Internal  potential  of  a  system  divided  into  a  certain 
number  of  homogeneous  phases.  —  More  generally,  let  us  consider 
a  system  formed  by  a  certain  number  of  homogeneous  phases  1, 
2,  .  .  .  ,  <£;   let  MI,  M2,  .  .  .  ,  M9  be  the  masses  of  these  phases 
and  S,  S',    ...  the  areas  of  the  surfaces  limiting  these  various 
phases  or  which  separate  them  from  each  other;    the  internal 
potential  of  the  system  will  be  of  the  following  form  : 

(1)  &=Ml&l+Mfrt+  .  .  .M,fif+AS+A'S'+  .  .  . 

fFj  is  a  quantity  which  depends  upon  the  temperature,  the  nature, 
the  state  and  density  of  the  substance  1  ;  SF2,  .  .  .  ,  £F^  depend 
in  an  analogous  manner  upon  the  substances  2,  .  .  .  ,  <j>;  as  to 
A,  it  is  a  quantity  depending  upon  the  temperature,  nature,  state, 
and  density  of  the  substance  bounded  by  the  surface  S  or  of  the 
substances  which  it  separates;  the  quantities  A',  .  .  .  have  analo- 
gous properties. 

272.  Comparison  with  the  form  used  in  the  preceding  chapters. 
—  The  laws  developed  from  the  sixth  chapter  on  do  not  follow  from 
an  employment  of  formula  (1),  but,  as  we  have  seen  in  Art.  89, 
from  the  use  of  the  simpler  expression 

(2)  S=Ml*l+Mf,+  .  .  .  +M&+, 

which  is  deduced  from  formula  (1)  by  neglecting  the  terms  AS 
A'S',  .  .  . 


364  THERMODYNAMICS  AND  CHEMISTRY. 

It  is  then  possible  that  the  laws  deduced  from  this  simplified 
formula  may  be  found  inexact  for  certain  cases;  a  comparison 
will  emphasize  the  importance  of  the  error  which  may  be  com- 
mitted by  neglecting  the  terms  A,S  A'S',  .  ,  ,  '• 

Suppose  it  is  desired  to  find  the  form  assumed  by  a  system 
composed  of  one  or  of  several  fluid  masses  under  the  action  of  their 
weight.  If  the  simplified  formula  (2)  is  taken  as  starting-point, 
propositions  are  arrived  at  which  are  the  laws  of  elementary  hy- 
drostatics; compared  with  observation,  these  laws  are  shown  to 
be  contradicted  by  a  great  number  of  phenomena  called  the  capil- 
lary phenomena;  to  take  account  of  these  phenomena,  it  suffices 
to  take  no  longer  as  starting-point  the  simplified  equation  (2), 
but  the  complete  equation  (1). 

This  comparison  leads  us  quite  naturally  to  put  the  following 
question : 

In  what  cases  is  it  permissible,  in  chemical  mechanics,  to  make 
use  of  the  simplified  formula  (2)  ?  In  what  cases,  on  the  contrary, 
is  it  necessary  to  employ  the  complete  formula  (1)? 

273.  When  all  the  phases  are  of  very  great  mass,  the  theories 
developed  in  the  preceding  chapters  are  exact. — The  reply  to 
this  question  depends  upon  an  essential  remark,  namely: 

When  all  the  dimensions  of  a  system  are  multiplied  by  the 
same  number,  the  different  masses  composing  the  system  are  mul- 
tiplied by  the  cube  of  this  number,  while  the  areas  of  the  surfaces 
limiting  or  separating  the  various  bodies  of  this  system  are  mul- 
tiplied only  by  the  square  of  this  number;  if,  therefore,  the  system 
is  increased  in  size,  the  various  masses  composing  it  will  increase 
much  more  rapidly  than  the  surfaces  met  with ;  if,  on  the  con- 
trary, the  dimensions  of  the  system  are  reduced,  the  various  masses 
composing  it  will  diminish  much  more  rapidly  than  the  surfaces 
of  separation. 

From  this  the  following  consequence:  One  may  always  attrib- 
ute to  the  various  phases  which  compose  a  system  masses  great 
enough  so  that  the  terms  AS,  A'S'  in  formula  (1)  are  negligible 
compared  with  the  term  (M151  +  M2$2  +  .  .  .  Af^Sfy);  one  may 
then  make  use  of  the  simplified  formula  (2),  whence  follow  all  the 
laws  developed  in  what  precedes ;  in  other  words,  the  laws  of  chem- 
ical mechanics  developed  in  the  preceding  chapters  are  exact  whenever 


CAPILLARY  ACTIONS-APPARENT  FALSE  EQUILIBRIA.  365 

the  various  phases  into  which  the  system  studied  is  divided  have 
sufficiently  great  masses. 

When,  on  the  contrary,  one  or  several  of  these  phases  have 
very  small  masses  these  laws  may  be  in  default. 

274.  Application   to   the  vaporization  of  a  liquid;    case  in 
which  the  classic  theory  is  exact. — Let  us  take,  for  example,  the 
phenomenon  of  the  vaporization  of  water. 

A  mass  Ml  of  water  is  in  contact  with  a  mass  M2  of  vapor  across 
a  surface  21;  Sl  and  S2  are  the  surfaces  which  complete,  with  the 
surface  2,  the  boundaries  of  the  masses  Mt  and  M2;  the  internal 
thermodynamic  potential  of  the  system  is  of  the  form 

(3)  5=M151+M252+aI+A1Sl+A2S2. 

Suppose,  in  the  first  place,  the  masses  of  liquid  and  vapor  are 
both  very  great ;  in  this  case,  from  the  preceding  remark,  the  terms 
of  the  potential  F  which  are  proportional  to  these  masses  are 
very  great  compared  with  the  terms  proportional  to  the  areas  of 
the  limiting  surfaces;  in  the  expression  for  the  internal  thermo- 
dynamic potential  one  may  neglect  these  last  and  reduce  the 
expression  (3)  to  the  simplified  form 

(4)  ^f^M^+Mffy 

Now  this  simplified  form  is  the  same  one  deduced  from  the 
classic  theory  of  evaporation;  hence  the  same  laws  comprised  in 
this  theory  are  again  found : 

At  every  temperature  there  exists  but  one  pressure  for  which 
the  liquid  remains  in  equilibrium  in  contact  with  the  vapor;  under 
a  pressure  less  than  this  tension  of  saturated  vapor  the  liquid 
vaporizes;  under  a  higher  pressure  the  vapor  condenses. 

But  these  laws,  which  the  classic  theory  regards  as  general, 
appear  to  us  here  as  subordinated  to  one  condition:  it  is  that  the 
masses  of  liquid  and  vapor  considered  are  always  great  masses. 
In  all  cases  where  this  condition  is  not  fulfilled  we  may,  without 
contradiction,  find  these  laws  inexact. 

275.  Case  where  the  liquid  contains  a  very  small  vapor  bub- 
ble.    Theory  of  retardation  of  ebullition. — Suppose,  for  instance, 
a  small  vapor  bubble  be  surrounded  by  liquid ;  we  may  no  longer 


366  THERMODYNAMICS  AND  CHEMISTRY. 

in  equation  (3)  neglect  the  term  al  as  compared  with 
these  two  terms  may  be  of  the  same  order  of  greatness,  and  even, 
if  the  bubble  is  infinitely  small,  the  absolute  value  of  the  term  a2 
will  be  infinitely  great  compared  with  the  absolute  value  of  the 
term  M2$2;  the  presence  of  the  term  a.2  in  the  expression  for  the 
internal  thermodynamic  potential  will  change  entirely  the  con- 
clusions which  may  be  drawn  from  the  study  of  this  potential; 
so  that  the  laws  of  equilibrium  of  a  very  small  bubble  of  vapor 
within  a  liquid  may  be  entirely  different  from  the  laws  of  equilib- 
rium of  a  great  mass  of  vapor  in  contact  with  a  great  mass  of  liquid. 

These  laws  of  the  equilibrium  of  a  small  bubble  of  vapor  within 
a  great  mass  of  liquid  may  be  established  in  detail  by  means  of 
the  principles  we  have  just  exposed;  they  lead  to  the  following 
consequences : 

In  order  that  a  bubble  of  vapor  may  increase  at  the  expense 
of  the  surrounding  liquid,  it  is  not  sufficient  for  the  pressure  at  a 
point  near  this  bubble  to  be  less  than  the  tension  of  saturated 
vapor;  it  is  further  necessary  that  the  radius  of  the  bubble  be 
greater  than  a  certain  limit,  a  limit  depending  upon  the  tempera- 
ture and  the  pressure;  when  the  radius  of  the  bubble  is  less  than 
this  limit,  not  only  the  bubble  cannot  grow  in  size  at  the  expense 
of  the  surrounding  liquid,  but  further,  the  vapor  which  it  encloses 
necessarily  condenses;  the  bubble  collapses. 

From  this,  a  bubble  of  vapor  will  never  be  formed  in  a  region 
where  the  liquid  is  continuous;  in  fact,  if  such  a  bubble  could 
begin  to  form,  its  radius  would  be  at  first  infinitely  small,  less  than 
the  limiting  radius  of  which  we  have  spoken;  whence,  instead  of 
continuing  to  grow,  it  would  collapse. 

We  see  that  boiling  can  never  commence  but  at  those  points 
where  gaseous  bubbles  of  a  certain  size  already  exist;  this  is 
actually  the  conclusion  drawn  from  numerous  and  precise  observa- 
tions made  on  the  retardation  of  ebullition  by  Donny,  Dufour, 
and  Gernez. 

276.  Generalization  of  the  preceding  considerations. — What 
we  have  just  said  on  the  subject  of  the  transformation  of  a  liquid 
into  vapor  may  be  generalized  without  difficulty,  and  we  are  thus 
led  to  the  following  conclusions: 

When  a  certain  substance  a  may  be  formed  at  the  expense  of 


CAPILLARY  ACTIONS— APPARENT  FALSE  EQUILIBRIA.  367 

another  substance  b,  the  conditions  which  permit  predicting  if 
the  transfer  will  take  place  or  will  not  take  place  are  quite  different 
according  as  a  mass  a  of  considerable  extent  exists  beforehand  in 
contact  with  the  substance  6  or  if  the  substance  b  exists  alone  at 
the  beginning  of  the  modification. 

It  is  in  the  first  case  only  that  the  consequences  habitually 
deduced  from  the  principles  of  thermodynamics  are  legitimate; 
they  are  not  applicable  to  the  second  case;  if,  for  example,  to 
the  number  of  these  consequences  is  added  a  proposition  affirming 
that,  in  certain  conditions,  a  considerable  mass  of  the  substance 
a,  put  in  contact  with  the  substance  6,  will  increase  at  the  expense 
of  this  substance,  one  could  not  conclude  that  the  substance  a 
would  be  formed  within  the  substance  6,  originally  homogeneous. 

277.  Various  phenomena  explained  by  these  consequences. — 
These  considerations  do  not  apply  merely  to  the  retardation  of 
boiling;  they  completely  explain  a  great  number  of  phenomena: 

The  retardiruj  of  condensation  of  a  vapor  compressed  beyond 
the  tension  of  saturated  vapor,  retardation  to  which  an  end  is 
put  by  the  introduction  of  small  liquid  drops  or  of  solid  dust; 

The  supersaturation  of  gaseous  solutions,  which  ceases  by  the 
introduction  of  a  bubble  of  gas; 

The  retardation  of  decomposition  of  certain  endothermic  sub- 
stances (oxygenated  water/ nitrous  acid),  retardation  ceasing  by 
the  introduction  of  gaseous  bubbles  or  of  porous  substances  con- 
taining gas; 

The  undercooling  of  a  liquid,  which  ceases  by  the  introduction 
of  a  particle  of  the  solid  to  be  produced; 

The  supersaturation  of  a  salt  solution,  to  which  an  end  is  put 
by  dropping  in  a  crystal  of  the  salt  to  precipitate  or  of  an  iso- 
morphous  salt; 

The  retarding  of  transformation  of  one  crystalline  form  into 
another;  for  instance,  the  retardation  of  transformation  of  clino- 
rhombic  sulphur  into  orthorhombic  sulphur  at  ordinary  tempera- 
ture, retardation  stopped  by  contact  with  a  piece  of  rhombic 
sulphur;  the  retardation  of  transformation  of  orthorhombic 
sulphur  into  clinorhombic  sulphur,  at  temperatures  above  97°.2, 
retardation  ceasing  by  contact  with  a  clinorhombic  particle. 


368  THERMODYNAMICS  AND  CHEMISTRY. 

278.  These  phenomena  represent  apparent  false  equilibria. — 

In  no  one  of  the  cases  we  have  just  cited  is  there  produced,  properly 
speaking,  states  of  false  equilibrium;  all  the  equilibrium  states 
experiment  reveals  are  predicted  by  the  principles  of  thermo- 
dynamics, provided  that,  in  applying  these  principles,  use  is  made 
of  the  complete  equations  where  account  is  taken  of  the  terms 
proportional  to  the  surfaces  of  contact  of  the  various  phases;  if 
there  seems  to  be  contradiction  in  certain  cases  between  observa- 
tion and  theory,  it  is  because  the  theory  has  been  simplified  by 
means  of  an  unwarranted  supposition;  in  all  the  cases  of  which 
we  have  just  spoken  there  are  produced  only  apparent  false  equi- 
libria. 


CHAPTER  XVIII. 
GENUINE  FALSE  EQUILIBRIA. 

279.  Genuine  false  equilibria  exist.  Investigations  of  H. 
Pelabon  on  the  formation  of  sulphuretted  hydrogen. — The  false 
equilibria  studied  in  the  preceding  chapter  are  apparent  false 
equilibria;  they  are  nowise  in  disaccord  with  the  principles  of 
thermodynamics ;  they  contradict  merely  an  additional  hypothesis 
which  represents,  in  certain  cases,  a  sufficient  approximation 
and  which,  in  other  cases,  cannot  be  conserved. 

Must  we  conclude  that  all  the  false  equilibria  are  apparent  falst 
equilibria?  Does  observation  never  show  any  case  of  equilibrium 
irreconcilable  with  the  principles  of  thermodynamics?  Certain 
authors  seem  to  have  thought  so ;  but  we  do  not  think  their  opinion 
can  be  accepted  on  this  point. 

Let  us  analyze  the  fo  lowing  observation,  which  is  due  to  H. 
Pelabon:1 

Several  glass  tubes,  containing  0.02  gr.  of  pure  sulphur  and  of 
pure  hydrogen,  were  placed  in  a  furnace  whose  temperature  oscil- 
lated between  280°  and  285°.  After  six  hours'  heating  the  gases 
in  these  two  tubes  were  analyzed  after  sudden  cooling;  denoting 
by  V  the  volume,  in  the  normal  conditions  of  temperature  and 
pressure,  of  the  gas  contained  in  the  tube,  by  v  .the  volume  after 
absorption  of  the  sulphuretted  hydrogen  by  potash,  and  by  p 
the  ratio  of  the  partial  sulphuretted  hydrogen  pressure  in  the 
gaseous  mixture  to  the  total  pressure  of  this  latter,  there  resulted: 

V = 8.766  cm.        v = 8.547  cm.        p = 0.025 
y=10.2  v=9.95  ^=0.0248 

1  H.  PELABON,  Memoires  de  la  Societe  des  Sciences  physiques  et  naturettes 
de  Bordeaux,  5th  S.,  v.  3,  p.  257,  1898. 


370  THERMODYNAMICS  AND  CHEMISTRY. 

After  38  hours  the  gas  in  the  other  tube  was  analyzed  and  gave 
V  =  8.76  cm.  v  =  7.9  cm.  p = 0.098. 

After  162  hours  of  heating: 

7  =  7.135  cm.          v=4.75cm.          ,0=0.3356 

After  300  hours: 

7=9.25  cm.          v=6.15cm.          ,0=0.3354 

This  shows  that  the  ratio  p  increases  at  first  with  the  duration 
of  heating;  but  after  160  hours  the  ratio  p  attains  a  value  which 
it  afterwards  keeps  indefinitely,  if  the  temperature  does  not 
change;  when  p  reaches  this  value  equilibrium  is  reestablished 
in  the  system. 

One  would  expect,  from  the  laws  of  thermodynamics,  that  a 
system  containing  sulphur,  hydrogen,  and  hydrogen  sulphide,  where 
the  ratio  p  has  a  value  greater  than  this  limit  0.3355,  should  be  the 
seat  of  a  partial  decomposition  of  hydrogen  sulphide  when  it  is 
kept  at  285°;  one  would  then  see  the  ratio  p  diminish  as  the  time 
of  heating  was  increased,  and  approach  the  same  limit  0.3355. 
This  is  not  so;  however  rich  in  sulphuretted  oxygen  is  the  gaseous 
mixture  submitted  to  the  temperature  of  280°,  this  sulphuretted 
hydrogen  remains  unaltered,  and  that  even  if  the  tube  encloses 
only  sulphur  and  hydrogen  sulphide  gas  without  admixture  of 
hydrogen.  At  the  temperature  of  280°,  in  a  system  containing 
hydrogen,  hydrogen  sulphide,  and  saturated  sulphur  vapor,1  equi- 
librium is  established  every  time  the  value  of  the  ratio  p  equals 
or  surpasses  0.3355. 

Does  the  value  p= 0.3355  correspond  to  a  veritable  equilibrium 
state  for  the  temperature  280°?  Sulphuretted  hydrogen  being  a 
compound  strongly  exothermic,  the  value  of  p  corresponding  to 
a  veritable  equilibrium  state  should  diminish  as  the  temperature 
rises  (Art.  174) ;  at  the  temperature  of  440°  the  system  studied  is 
in  a  state  of  incontestable  true  equilibrium,  and  this  state  corre- 
sponds to  a  value  of  p  included  between  97.5  and  98.2;  at  the 

1  P&LABON  has  shown  that  liquid  sulphur  absorbs  hydrogen  sulphide  in 
abundance;  this  circumstance  complicates  somewhat  the  verification  of  the 
preceding  laws,  as  may  be  seen  in  PELABON'S  memoir,  I.  c. 


GENUINE  FALSE  EQUILIBRIA.  371 

temperature  of  280°  the  value  of  p  which  would  correspond  to  a 
state  of  veritable  equilibrium  would  differ  very  slightly  from  1. 

We  may  therefore  state  the  following  proposition: 

At  280°,  as  long  as  p  is  included  between  0  and  0.3355,  there  s 
formed  hydrogen  sulphide  gas.  reaction  conforming  with  the  pre- 
dictions of  thermodynamics;  when  p  is  included  between  0.3355 
and  1  the  system  is  in  equilibrium,  although,  according  to  the  pre- 
dictions of  thermodynamics,  there  should  be  hydrogen  sulphide 
gas  formed;  in  this  last  case  the  system  is  in  the  state  of  false 
equilibrium. 

Is  the  state  we  have  just  defined  merely  one  of  apparent  false 
equilibrium?  It  does  not  seem  that  we  may,  in  any  manner, 
apply  to  it  the  considerations  which  allowed  us  to  reduce 
to  thermodynamical  laws  the  retardations  of  boiling  and  the 
analogous  phenomena. 

Is  this  a  state  of  illusory  equilibrium?  May  we  not  admit 
that  the  sulphuretted  hydrogen  continues  to  be  formed  in  the 
mixture  kept  at  280°  and  in  which  p  has  a  value  greater  than 
0.3355,  but  formed  so  slowly  that  this  reaction  escapes  all  control? 
This  is  an  opinion  which  may  be  admitted,  that  observation  evi- 
dently cannot  prove  false,  but  which  it  cannot  any  more  prove 
true.1  It  seems  to  us  simpler  and  more  logical  to  admit  that  a 
system  for  which  the  ratio  p  exceeds  0.3355  remains  really  in 
equilibrium  at  the  temperature  of  280°,  that  such  a  state  of  equi- 
librium is  incompatible  with  the  laws  of  thermodynamics,  and 
that  the  latter  have  need  of  being  modified  and  extended  so  as  to 
take  into  account  states  of  false  equilibrium.2 

1  Thus  Van't  Hoff  thinks  that  in  certain  cases  even  the  geological  periods 
are  insufficient  for  the  state  of  veritable  equilibrium  to  be  attained  (Archives 
neerlandaises  des  Sciences  exactes  et  naturettes,  2d  S.,  v.  6,  p.  489,  1901). 

'According  to  Max  Bodenstein  (Zeitschrift  filr  physikalische  Chemie, 
v.  29,  pp.  147,  295,  315;  1899)  all  the  effects  observed  by  Pelabon  on  the 
formation  of  sulphydric  acid,  by  Ditte  and  by  Pelabon  on  the  formation 
of  selenhydric  acid,  by  A.  Gautier  and  Helier  on  the  combination  of 
oxygen  and  hydrogen  would  be  disputed;  there  would  be  produced  false 
equilibria  in  none  of  these  cases.  It  seems  to  me  difficult  to  accept  the 
affirmations  of  this  author,  whose  researches  appear  to  have  been  made  in 
a  very  hasty  way,  who  is  in  contradiction  with  all  his  predecessors,  even 
in  the  cases  where  the  latter  have  encountered  states  of  veritable  equilibrium, 


372 


THERMODYNAMICS  AND  CHEMISTRY. 


280.  The  condition  of  false  equilibrium  is  not  expressed  by 
an  equality. — When  the  laws  of  false  equilibrium  are  compared 
with  the  laws  governing  the  states  of  true  equilibrium,  a  first 
difference  immediately  attracts  attention:    a  law  of  true  equi- 
librium is  expressed  by  an  equality;   we  have  seen  numerous  ex- 
amples of  this  in  our  work;   on  the  contrary,  a  law  of  false  equi- 
librium is  expressed  by  an  inequality.     Thus,  in  the  preceding 
case,  at  the  temperature  of  280°  the  system  is  in  equilibrium  if 
p  is  at  least  equal  to  0.3355. 

281.  Region  of  false  equilibria.     Boundary  line  of  false  equi- 
libria.— On   the  two    coordinate  axes  OT,  Op  (Fig.  115)  lay  off 


A' 


O  T  T  T 

FIG.  115. 

the  temperatures  as  abscissae  and  the  values  of  p  as  ordinates; 
let  L  be  the  point  whose  abscissa  represents  280°  C.  and  whose 
ordinate  has  the  value  ,0  =  0.3355;  every  point  C  located  below 
L  on  the  straight  line  TL  represents  a  state  of  the  system  where 
hydrogen  combines  with  sulphur  to  form  hydrogen  sulphide;  on 
the  contrary,  every  point  E  situated  above  L  on  the  line  TL 
represents  a  state  where  the  system  remains  in  false  equilibrium. 

When  the  temperature  T  is  varied  the  point  L  varies;  thus, 
according  to  the  observations  of  Pelabon,  we  have,  as  coordinates 
of  the  point  L: 

who  gives  no  plausible  explanation  of  these  contradictions,  and  finally  who 
has  read  in  a  very  superficial  manner  the  writings  whose  conclusions  he 
contests.  (See  P.  DUHEM,  Zeitschrift  fur  physikalische  Chemie,  v.  29,  p.  711, 
1899.) 


GENUINE  FALSE  EQUILIBRIA.  373 

T  P 

200°  C.  0.0210 

235  .0541 

255  . 13 

280-285  .3355 

310  .69 

350  .972 

As  the  temperature  T  increases,  the  point  L  describes  a  line  LU 
which  rises  rapidly  from  left  to  right. 

This  line  divides  the  plane  into  two  regions;  every  point  of 
the  region  situated  below  the  line  LU  represents  a  state  where 
the  system  is  the  seat  of  a  combination;  it  is  the  region  of  com- 
bination; every  point  of  the  region  situated  above  the  line  LU 
represents  a  state  of  false  equilibrium ;  it  is  the  region  of  false  equi- 
libria; the  line  LU  is  the  boundary  of  the  false  equilibria. 

When  a  tube  containing  0.02  gr.  of  sulphur  per  cubic  centi- 
metre in  an  atmosphere  of  hydrogen  is  brought  to  a  temperature 
T,  the  proportion  p  in  hydrogen  sulphide  approaches  the  ordinate 
corresponding  to  the  boundary  of  the  false  equilibria  and  never 
goes  beyond  it;  thi?  limiting  value  of  p  is  furthermore  attained 
the  more  rapidly  when  the  temperature  T  is  higher. 

If  for  each  temperature  T  the  number  h  of  hours  of  heating 
are  laid  off  as  abscissae  and  the  values  of  p  as  ordinates,  a  curve  is 
obtained  (Fig.  116)  which  rises  at  first  from  left  to  right,  then 


310 


280° 


Q6    38  162  300  ft 

FIG.  116. 

becomes  parallel  to  Oh',  for  the  first  part  of  this  curve  the  rise 
is  the  more  rapid  as  the  temperature  rises;  at  temperatures  above 
350°  equilibrium  is  reached  in  a  few  minutes. 


374 


THERMODYNAMICS  AND  CHEMISTRY. 


282.  Case  in  which  the  region  of  false  equilibria  separates 
two  regions  corresponding  to  two  reactions  the  inverse  of  each 
other.  Work  of  Jouniaux  on  the  reduction  of  silver  chloride  by 
hydrogen.  —  In  the  case  just  spoken  of,  the  boundary  of  the  false 
equilibria  is  single;  it  separates  the  region  of  false  equilibria  from 
a  region  where  a  reaction  of  well-determined  direction  is  produced, 
a  combination. 

In  other  cases  the  condition  in  order  that  the  system  be  in 
the  state  of  false  equilibrium  is  expressed  by  a  double  inequality; 
the  region  of  false  equilibria  is  comprised  between  two  bounda- 
ries; one  of  them  separates  this  region  from  one  where  a  certain 
reaction  is  produced;  the  other,  from  a  region  where  the  inverse 
reaction  is  produced. 

The  investigations  of  A.  Jouniaux  l  on  the  two  reactions,  the 
inverse  of  each  other, 


Ag+HCl=AgCl+H, 

will  give  us  an  example. 

The  composition  of  a  definite  volume  of  a  mixture  of  hydrogen 
and  of  hydrochloric  acid  gas  is  represented,  in  Jouniaux's  researches, 
by  the  ratio  p  of  the  volume  of  hydrochloric  acid  gas  to  the  total 
volume  of  the  mixture,  these  volumes  being  read  in  the  same 
pressure  and  temperature  conditions. 

At  ordinary  temperature  and  under  a  pressure  of  380  millimetres 
of  mercury  a  tube  of  Jena  glass  containing  silver  chloride  is 
filled  with  pure,  dry  hydrogen;  the  tube  is  brought  to  448°;  after 
being  heated  a  time  h  it  is  suddenly  cooled  and  the  gaseous  mix- 
ture it  contains  is  analyzed,  with  the  following  results  : 


h 

P 

h 

p 

7  hours 
24      " 
36      " 

0.7109 
0.8157 
0.8246 

70  hours 
408      " 
504      " 

0.8866 
0.8888 
0.8842 

1  A.  JOUNIAUX,  Comptes  Rendus,  v.  129,  p.  883,  1889;  Actions  des  hydra- 
tides  halogenes  sur  I'argent  et  reactions  inverses,  Lille  Thesis,  1901. 


GENUINE  FALSE  EQUILIBRIA. 


375 


If  the  values  of  h  are  laid  off  as  abscissae  and  of  p  as  ordinates, 
a  curve  oof  is  obtained  which  rises  at  first  from  left  to  right,  then 
becomes  parallel  to  Oh  (Fig.  117).  After  about  60  hours'  heating 


i.o 


A' 


a 


.  Ji 


2     4     6     8    10  12   14    16    18  20  22  days 

FIG.  117. 

the  reduction  of  silver  chloride  by  hydrogen  stops ;  the  value  of  p 
is  then  close  to 

r= 0.8888. 

The  system  is  in  equilibrium. 

If  the  hydrogen  introduced  into  the  tube  were  employed 
entirely  to  reduce  silver  chloride,  the  hydrochloric  acid  set  free 
would  exert  a  pressure  exactly  double  that  exerted  by  the  hydrogen 
absorbed ;  the  inverse  system  of  this  one  we  have  taken  as  starting- 
point  in  the  preceding  experiment  is  therefore  formed  of  silver  in 
the  presence  of  hydrochloric  acid,  this  latter  exercising,  at  the 
ordinary  temperature,  a  pressure  measured  by  760  millimetres  of 
mercury.  Let  us  take  such  a  system ;  heat  it  to  448°  for  a  time  h, 
which  we  lay  off  as  abscissae,  and  lay  off  the  value  p  as  ordinates; 
the  following  results  are  obtained: 


h 

p 

h 

p 

8  hours 
24      " 
36      " 

0.9598 
0.9392 
0.9292 

70  hours 
408      " 
504      " 

0.9158 
0.9167 
0.9155 

376 


THERMODYNAMICS  AND  CHEMISTRY. 


These  results  are  represented  by  the  curve  AA' ,  which  descends 
at  first  from  left  to  right,  then  becomes  parallel  to  Oh.  After 
60  or  70  hours  the  formation  of  silver  chloride  stops;  the  value 
of  p  is  sensibly 

#=0.9155, 

and  the  system  is  then  in  equilibrium. 

Thus  at  448°  a  system  of  same  percentage  composition  as  one 
of  the  two  inverse  systems  studied  in  what  precedes  will  be  in  a 
state  of  false  equi  ibrium  whenever 

r<p<R. 

It  is  here  a  double  inequality  which  defines  the  condition  of 
equilibrium. 

The  two  values  of  r,  Rf  always  determined  by  taking  at  ordi- 
nary temperature  hydrogen  under  a  pressure  of  380  millimetres 
of  mercury  and  hydrochloric  acid  gas  at  760  millimetres,  vary 
with  the  temperature  T  to  which  the  system  is  brought.  The 
values  obtained  by  Jouniaux  are  the  following: 


T 

r 

R 

200° 
250 
350 
448 
490 

hardly  appreciable 
0.05 
0.7588 
0.8888 
0.9036 

1 
1 
0.95 
0.9155 
0.9094 

If  the  values  of  T  are  laid  off  as  abscissae  and  of  r  and  R  as 
ordinates,  two  boundaries  IV  and  LL'  are  obtained  (Fig.  119,  p.  380). 
Between  these  two  lines  extends  the  region  of  false  equilibria;  the 
line  IV  separates  this  region  from  that  where  hydrogen  reduces  sil- 
ver chloride;  the  line  LL'  separates  this  region  from  that  where 
hydrochloric  acid  attacks  silver. 

283.  Another  example:  Carbonate  of  magnesium  and  bicar- 
bonate of  potassium.  Engel's  studies. — Here  is  another  example  * 


1  ENGEL,  Comptes  Rendus,  v.  101,  p.  749,  1885. 


GENUINE  FALSE  EQUILIBRIA.  377 

where  the  region  of  false  equilibria  separates  two  regions  which 
correspond  respectively  to  two  reactions  the  inverse  of  each  other. 
Carbonate  of  magnesium  combines  with  bicarbonate  of  potas- 
sium, forming  the  double  salt  of  formula 

CO3Mg,CO3nK-f4H2O. 

Put  in  the  presence  of  water,  this  salt  decomposes;  the  carbon- 
ate of  magnesium,  almost  insoluble,  is  deposited  while  the  potas- 
sium bicarbonate  is  dissolved;  when  the  solution  of  potassium 
bicarbonate  is  sufficiently  concentrated,  the  decomposition  stops 
and  equilibrium  is  established. 

At  this  moment  the  system  is  divided  into  three  phases:  the 
solution,  the  double  salt,  the  solid  -carbonate  of  magnesium;  it 
is  besides  formed  of  three  independent  components,  bicarbonate 
of  potassium,  magnesium  carbonate,  water;  it  is  therefore  a  bi- 
variant  system:  under  atmospheric  pressure  there  should  corre- 
spond to  every  temperature  a  state  of  true  equilibrium  defined 
by  a  given  composition  of  the  solution.  The  solution  containing 
almost  exclusively  potassium  bicarbonate  and  water,  its  com- 
position may  be  fixed  by  its  concentration  s.  The  equilibrium 
would  then  correspond,  for  each  temperature  T,  to  a  value  S  of 
the  concentration;  at  this  temperature,  in  presence  of  a  solution 
of  concentration  less  than  S(s<S),  the  double  salt  would  decom- 
pose; on  the  contrary,  in  presence  of  a  solution  of  concentration 
greater  than  S(s>S).  the  bicarbonate  of  potassium  would  com- 
bine with  the  magnesium  carbonate. 

This  is  not  at  all  how  things  actually  happen. 

At  a  given  temperature  T  the  solution  decomposes  the  double 
salt  while  the  concentration  s  is  less  than  a  certain  given  limit 
o(s<a);  the  concentration  being  comprised  between  two  limits 
a,  2,  the  second  higher  than  the  first, 


the  system  remains  in  eqidlibrium;  finally,  when  the  concentra- 
tion exceeds  I(*>S)  the  potassium  bicarbonate  combines  with 
the  magnesium  carbonate. 


378 


THERMODYNAMICS  AND  CHEMISTRY. 


Combination 


(Decomposition 


T 
FIG.  118. 


On  the  two  axes  of  rectangular  coordinates  OT,  Os  (Fig.  118) 

lay  off  the  temperatures  T  as  ab- 
scissae, and  the  concentrations  as 
ordinates;  for  the  same  tempera- 
ture T  let  I  and  L  be  the  points 
having  the  ordinates  a  and  I  re- 
spectively. 

Every  point  D  located  below 
the  point  I  on  the  straight  line  Tl 
represents  a  state  of  the  system 
within  which  the  double  salt  de- 
composes; every  point  C  situated 
above  L  represents  a  state  in  which 
potassium  bicarbonate  combines 
with  magnesium  carbonate;  any  point  E  included  between  /  and 
L  represents  a  state  of  false  equilibrium. 

When  the  temperature  T  varies,  the  concentrations  <r,  I  vary 
likewise  and  the  two  points  I,  L  describe  two  lines  llr,  LI/;  accord- 
ing to  Engel's  observations,  at  temperatures  included  between 
14°  and  40°  these  lines  both  rise  from  left  to  right.  They  divide 
the  plane  into  three  regions;  the  region  situated  below  the  line  II' 
is  the  region  of  decomposition;  the  region  above  the  line  LL'  is  the 
region  of  combination;  the  one  between  the  lines  II'  and  LL'  is  the 
region  of  false  equilibria. 

*  284.  Return  to  the  idea  of  reversible  modification. — The 
chemical  equilibria  studied  in  the  preceding  chapters  all  had  a 
common  property;  each  of  these  equilibria  was  the  common  limit 
of  two  reactions  the  inverse  of  each  other  (Arts.  46,  47,  53,  54,  55, 
56).  From  this  property  a  consequence  of  capital  importance 
followed;  a  continuous  series  of  equilibrium  states  constituted  a 
reversible  transformation  (Art.  59,  60,  61).  Now,  it  is  because 
a  continuous  series  of  equilibrium  states  constituted  a  reversible 
transformation  that  it  was  possible  to  apply  to  these  states  of 
equilibrium  the  various  theorems  of  thermodynamics. 

All  these  theorems,  all  the  corollaries  which  may  be  deduced 
from  them,  become  useless  when  the  chemical  equilibrium  is  no 
longer  the  common  limit  of  two  reactions  the  inverse  of  each  other. 
We  must  not  be  astonished,  therefore,  to  find  the  states  of  false 


GENUINE  FALSE  EQUILIBRIA.  379 

equilibrium  in  contradiction  with  propositions  such  as  the  phase 
rule  or  the  laws  of  the  displacement  of  equilibrium. 

These  contradictions  are  actually  met  with  at  every  instant; 
let  us  cite  one  as  example: 

The  combination  of  hydrogen  with  sulphur  is  exothermic. 
According  to  the  law  of  the  displacement  of  equilibrium  by  varia- 
tion of  temperature,  the  mass  of  hydrogen  sulphide  gas  formed 
within  a  system  where  hydrogen  and  sulphur  are  heated  at  con- 
stant volume  should  be,  at  the  instant  of  equilibrium,  the  feebler 
as  the  temperature  were  the  higher.  In  reality  (Art.  281)  the  pro- 
portion of  hydrogen  sulphide  gas  in  the  system,  at  the  moment 
the  reaction  ceases,  increases  indefinitely  with  the  temperature 
when  this  latter  is  raised  up  to  448°. 

285.  Relation  between  the  states  of  veritable  equilibrium  and 
the  states  of  false  equilibrium.  Action  of  hydrogen  on  silver 
chloride  and  the  inverse  action. — Often  a  chemical  system,  sus- 
ceptible of  possessing  states  of  false  equilibrium  at  certain  tem- 
peratures, may  have  states  of  veritable  equilibrium  at  other  tem- 
peratures, in  general  higher  than  the  first.  In  certain  cases  it  is 
possible  to  follow  the  continuous  passage  of  one  of  the  forms  of 
equilibrium  to  the  other. 

Take,  for  instance,  the  action  of  hydrogen  on  silver  chloride 
and  the  inverse  action  of  hydrochloric  acid  on  silver  (Art.  282). 
At  a  temperature  such  as  350°  or  448°  the  values  of  p  for  which 
the  system  can  be  in  equilibrium  are  included  between  two  limits 
r  and  R  which  are  notably  different;  but  as  the  temperature  rises 
these  two  limits  approach  each  other;  at  490°,  where  ris  hardly  less 
than  R,  we  had 

r= 0.9036,  #=0.9094. 

At  higher  temperatures1  the  two  limits  r  and  R  are  identical; 
the  same  equilibrium  state  is  reached  in  starting  from  the  system 
hydrogen-silver  chloride,  or  in  starting  from  the  system  hydro- 
chloric acid-silver.  This  common  value  of  r  and  R,  which  we 
shall  denote  by  (R,  is  the  following  at  various  temperatures: 

1  A.  JOUNIAUX,  Comptes  Rendus,  v.  132,  p.  1270,  1901 ;  Actions  des  hydra- 
cides  .  .  .  ,  Lille  Thesis,  1901. 


380 


THERMODYNAMICS  AND  CHEMISTRY. 


T 

(R 

T 

(R 

540° 

0.9155 

650° 

0.938 

600 

0.928 

700 

0.95 

In  place  of  the  two  boundaries  of  false  equilibria,  we  have  at 
temperatures  above  500°  but  a  single  line  of  veritable  equilibria, 
VV  (Fig.  119),  whose  every  point  has  for  abscissa  a  value  of  T 


1.0 

0.9- 

0.8 

0.7 

0.6- 

0.5- 

0.4- 

0.3 

0.2 

0.1 


False 
Equilibria 


0       100     300      300      400      500      600     700 

FIG.  119. 

and  for  ordinate  the  corresponding  value  of  (R.  This  line  rises 
from  left  to  right,  as  required  by  the  law  of  the  displacement  of 
equilibrium  by  variation  of  the  temperature,  for  the  reaction 

AgCl+H=Ag+HCl 

is  endothermic. 

286.  Action  of  hydrogen  on  selenium  and  the  inverse  action. 
Pelabon's  investigations. — The  relation  between  the  states  of 
false  equilibrium  and  the  states  of  veritable  equilibrium  is  more 
sharply  and  completely  brought  out  in  the  example  given  us  by 
the  dissociation  of  selenhydric  acid  and  the  inverse  action  of 


GENUINE  FALSE  EQUILIBRIA.  381 

selenium  on  hydrogen.  First  studied  by  Ditte,1  this  example  has 
been  the  object  of  researches  which  are  among  the  most  important 
of  physical  chemistry.  These  researches  are  due  to  Pelabon.2 

The  system  studied  encloses,  at  constant  volume,  liquid  sele- 
nium, vapors  of  selenium,  hydrogen,  and  selenhydric  acid. 

We  have  studied  (Chap.  XVI,  Art.  258)  the  condition  which 
controls  the  states  of  veritable  equilibrium  of  such  a  system.  We 
have  seen  that  if  p  denote  the  partial  hydrogen  pressure  in  the 
gaseous  mixture  and  p'  the  partial  pressure  of  selenhydric  acid, 
we  should  have,  within  a  system  in  veritable  equilibrium  at  the 
absolute  temperature  T, 

(1)  log£  =  ^+«logT+2, 

Pi          1 

m,  n,  and  z  being  three  constants  suitably  chosen. 

At  temperatures  above  350°  the  system  possesses  quite  char- 
acteristic states  of  veritable  equilibrium;  we  may  therefore,  by 
means  of  three  experiments  properly  chosen  and  performed  at 
temperatures  above  350°,  determine  the  values  which  should  be 
attributed,  in  the  preceding  equation,  to  m,  n,  z. 

These  values  once  determined,  we  may,  by  means  of  equa- 
tion (1),  calculate  for  each  absolute  temperature  T,  and  conse- 
quently for  each  centigrade  temperature  I,  the  value  that  ~ 

Y 

should  have  in  order  that  the  system  be  in  a  state  of  veritable 
equilibrium ;  take  two  rectangular  axes  and,  following  Pelabon, 
plot  the  centigrade  temperatures  as  abscissae  (Fig.  120),  and  as 

ordinates  the  values  of  the  ratio  p=  „  deduced  from  the  pre- 
ceding computation;  we  shall  obtain  the  curve  VJTV  which  will 
represent  the  states  of  veritable  equilibrium  of  the  system. 

At  a  temperature  higher  than  350°  the  system  will  be  in  equi- 
librium only  if  the  representative  point  is  on  the  curve  W]  if 

1  DITTE,  Annales  de  VEcole  normale  superieure,  2d  S.,  v.  i,  p.  293,  1872. 

2  PELABON,  Sur  la  dissociation  de  I'acide  selenhydrique  (Mem.  de  la  Soc. 
d.  Sciences  Phys.  et  Nat.  d.  Bordeaux,  5th  S.,  v.  3,  p.  141,  and  Paris,  A.  Her- 
mann, 1898). 


382 


THERMODYNAMICS  AND  CHEMISTRY. 


it  is  below  this  curve,  selenhydric  acid  will  be  formed  in  the  system; 
if  the  representative  point  is  above  this  curve,  the  selenhydric 
acid  contained  in  the  system  will  be  in  part  destroyed. 

It  will  be  quite  otherwise  for  temperatures  below  350°. 

Let  us  operate,  for  example,  at  270°. 

Take  tubes  which  contain  at  the  start  only  hydrogen  and 
selenium;  the  initial  value  of  p  is  equal  to  0;  heat  them  a  long  time 

A' 


r 
0.4 

0  1 

i 

^  

—  V 

\°j 

)ecoriip 

asition 

r 

X 

0.2 
0.1 

\l 

/ 

Cc 

mbinat 

Ion 

Talso 

Equilit 

^£ 

ria   j  . 
VV 

^ 

y}.... 

2 

O   100   200  CA300  4°0   500   600   700 

FIG.  120. 

at  270°;  p  increases  at  first  on  account  of  the  formation  of  selen- 
hydric acid;  then  after  a  sufficiently  long  time  equilibrium  is 
established  and  p  keeps  a  constant  value  which  is  nearly 

r= 0.048. 

Thus  in  an  experiment  where  the  tube  was  heated  for  490 
hours  p  had  the  value  0.0491;  in  another  in  which  the  tube  had 
been  heated  for  a  month  p  had  the  value  0.0478. 

Suppose,  on  the  contrary,  we  took,  at  the  beginning  of  the 
experiment,  mixtures  containing  a  large  proportion  of  selenhydric 
acid;  imagine,  for  instance,  that  the  initial  value  of  p  is  about 
0.40;  maintain  these  mixtures  at  270°;  the  selenhydric  acid  they 
contain  will  partly  decompose,  the  value  of  p  will  diminish;  after 
a  sufficient  time  of  heating  equilibrium  will  be  established  and 
p  will  then  keep  a  constant  value  near  to 

#=0.16. 

Thus  in  four  experiments  during  which  the  times  of  heating 
•were  respectively 


GENUINE  FALSE  EQUILIBRIA.  383 

192  hours,        288  hours,        480  hours,        490  hours, 
the  limiting  values  of  p  were  respectively  equal  to 

0.171,  0.165,  0.1605,  0.163. 

Therefore  every  time  p  verifies  the  inequality 

P<r 
at  270°,  hydrogen  combines  with  selenium;  whenever 

p>R 

at  270°,  selenhydric  acid  decomposes;   finally,  whenever  p  is  com- 
prised between  r  and  R, 

(2)  r<p<R 

at  270°,  the  system  is  in  equilibrium. 

If  one  computes,  from  formula  (1),  the  value  of  p  which,  at  the 
temperature  of  270°,  would  put  the  system  in  a  state  of  veritable 
equilibrium,  we  find  that  this  value  of  p  is  in  the  neighborhood  of 

(R  =  0.10. 

It  is  therefore  included  between  r  and  R  and  verifies  the  con- 
dition (2). 

Draw  a  parallel  AA'  (Fig.  120)  to  the  straight  line  Op,  having 
the  constant  abscissa  270°;  running  up  the  length  of  this  line 
from  A  to  A',  we  shall  meet  successively  a  point  /  of  ordinate  r, 
a  point  v  of  ordinate  (R,  and  a  point  L  of  ordinate  R.  The  points 
of  the  straight  line  AA'  located  below  the  point  I  represent 
systems  within  which  selenhydric  acid  is  formed;  the  points 
situated  between  I  and  L,  among  which  is  the  point  v,  represent 
systems  in  false  equilibrium;  the  points  situated  above  L  represent 
systems  in  which  selenhydric  acid  is  decomposed. 

When  the  temperature,  which  we  have  supposed  to  equal  270°, 
assumes  other  values,  the  points  Z  and  L  vary  and  describe  respec- 
tively the  lines  Cl  and  DL  (Fig.  120).  These  lines  divide  the  plane 
into  three  regions;  from  the  properties  possessed  by  a  system 
according  as  the  representative  point  is  in  one  or  another  of  these 
three  regions,  we  may  give  to  these  regions  the  following  denomi- 
nations: region  of  combi' nation,  located  below  the  line  Cl',  region 
of  false  equilibria,  situated  between  the  lines  Cl  and  DL;  region 
of  decomposition,  above  the  line  DL. 


384 


THERMODYNAMICS  AND  CHEMISTRY. 


The  line  V^y,  theoretical  prolongation  of  the  line  of  veritable 
equilibria,  determined  by  means  of  equation  (1),  is  located  wholly 
within  the  region  of  false  equilibria. 

The  line  Cl  leaves  the  temperature  axis  at  a  point  whose  ab- 
scissa is  about  250° ;  it  rises  continuously  from  left  to  right. 

The  line  DL  has  been  followed  by  Pelabon  from  the  tempera- 
ture 150°,  to  which  corresponds  a  value  of  R  closely  equal  to  0.3824; 
this  curve  descends  at  first  from  left  to  right  to  about  the  tem- 
perature of  270°,  where  it  possesses  a  minimum  ordinate  nearly 
equal  to  0.16;  then  it  rises  again  from  left  to  right. 

How  is  the  passage  made  from  the  law  of  formation  and  de- 
composition of  selenhydric  acid,  such  as  we  have  just  treated  it, 
to  the  law  which  controls  these  same  phenomena  at  temperatures 
above  350° ,  where  we  meet  no  more  false  equilibria  and  where 
a  simple  line  of  veritable  equilibria  W  separates  the  region  of  com- 
bination from  the  region  of  decomposition?  When  the  tempera- 
ture exceeds  300°,  the  three  lines  Cl,  DL,  Vv  approach  one  another; 
they  meet  at  about  325°,  and  remain  coincident  above  325°. 
Below  are  some  observations  due  to  Pelabon  which  show  this  effect: 


Temperatures. 

Duration 
of  Heating. 

r 

CR 

R 

300° 

212  hours 

0.124 

0.15 

0.172 

300 

322      " 

0.127 

0.15 

0.170 

315 

196      " 

0.164 

0.174 

0.185 

315 

320      " 

0.1625 

0.174 

0.180 

325 

175      " 

0.187 

0.192 

0.193 

325 

213      " 

0.1882 

0.192 

0.192 

287.  The  region  of  false  equilibria  separated  from  that  of 
veritable  equilibria  by  a  region  of  unlimited  reaction.  Action 
of  hydrogen  on  sulphur  and  the  inverse  action.— In  the  two 
cases  we  have  just  analyzed  there  is  passage  by  a  gradual 
transition  from  low  temperatures  where  the  system  admits  of  a 
region  of  false  equilibrium  which,  by  two  lines  of  limiting  false 
equilibria,  confines  inverse  reactions  to  two  regions,  to  high  tem- 
peratures where  the  system  admits  none  other  than  states  of  veri- 
table equilibrium. 

This  is  not  always  so. 

Let  us,  for  example,  reconsider  the  case  studied  in  Art.  279. 


GENUINE  FALSE  EQUILIBRIA. 


385 


Up  to  about  350°  the  hydrogen  sulphide  gas  is  not  decom- 
posable by  heat;  on  the  contrary,  above  200°  hydrogen  and 
sulphur  combine;  the  reaction  ceases  when  the  gaseous  mixture 
reaches  a  certain  content  of  hydrogen  sulphide;  this  content  is 
the  greater  as  the  temperature  is  the  higher. 

Therefore  from  200°  to  358°  a  limiting  line  of  false  equilibria 
tt'  (Fig.  121)  rises  from  left  to  right;  the  points  located  below  this 


P=l 


Combination 


200  300  400 

FIG.  121. 


500 


line  represent  states  of  the  system  such  that  hydrogen  and  sul- 
phur combine;  the  points  located  above  this  line  represent  states 
of  false  equilibria. 

At  temperatures  included  between  350°  and  400°  the  hydrogen 
sulphide  remains  undecomposable  by  heat;  in  return  hydrogen 
combines  in  totality  with  sulphur.  The  combination  is  unlimited. 
The  equilibrium  states  of  the  system  are  represented  by  the  part 
I'v  of  the  line  /o=l. 

When  the  temperature  assumes  a  value  greater  than  400°  the 
system  possesses  a  state  of  veritable  equilibrium,  common  limit  of 
these  two  inverse  reactions :  combination  of  hydrogen  and  sulphur, 
decomposition  of  hydrogen  sulphide.  At  440°,  for  instance,  the 
limit  obtained,  starting  either  from  the  pure  compound  or  from 
the  components,  is  the  same,  or,  at  least,  the  difference  is  of  the 
order  of  experimental  error. 

Two  tubes,  enclosing  each  0.02  gr.  of  sulphur  per  cubic  centi- 
metre, were  kept  for  6  hours  at  440°. 

The  first,  filled  with  hydrogen  sulphide,  gave  for  p  the  number 
0.975;  the  second,  which  contained  only  hydrogen  and  sulphur, 
gave  for  p  the  value  0.982,  practically  equal  to  the  preceding. 


386  THERMODYNAMICS  AND   CHEMISTRY. 

At  temperatures  above  400°  the  equilibrium  states  of  the 
system,  which  are  veritable  equilibrium  states,  are  represented 
by  the  line  w/ ;  the  region  of  combination  is  below  this  line,  the 
region  of  decomposition  above. 

The  formation  of  sulphuretted  hydrogen  being  exothermic  in 
the  conditions  indicated,  the  law  of  the  displacement  of  equi- 
librium by  variation  of  the  temperature  requires  the  line  vvr  to 
descend  from  left  to  right. 

It  is  clear,  from  these  important  observations  of  Pelabon,  that 
we  may  describe  in  the  following  way  the  influence  exerted  by  the 
temperature  on  the  formation  or  destruction  of  hydrogen  sulphide : 

At  temperatures  below  £  =  200°  hydrogen  sulphide  is  not 
decomposed;  hydrogen  does  not  act  upon  sulphur.  . ' 

Between  the  temperature  £  =  200°  and  the  temperature  r=350° 
hydrogen  sulphide  does  not  decompose;  hydrogen  combines  with 
sulphur  and  the  combination  is  limited;  the  gaseous  mixture 
obtained  is  the  richer  in  sulphuretted  hydrogen  as  the  temperature 
is  higher. 

At  temperatures  included  between  T=350°  and  0  =  400°  sul- 
phuretted hydrogen  is  decomposable  by  heat;  hydrogen  com- 
bines entirely  with  sulphur. 

Above  0=400°  hydrogen  sulphide  dissociates;  this  dissocia- 
tion is  limited  by  the  inverse  action,  and  it  is  the  more  marked 
as  the  temperature  is  higher. 

288.  Action  of  oxygen  on  hydrogen.  Work  of  A.  Gautier  and 
H.  Helier.1 — The  history  of  a  great  number  of  exothermic  com- 
binations appears  to  be  the  following: 

At  temperatures  less  than  t  the  compound  substance  is  indestruc- 
tible; the  elements  of  this  substance  cannot  combine. 

At  temperatures  comprised  between  t  and  r  the  compound  is 
indestructible;  the  elements  of  this  substance  may  combine  and  this 
combination  is  limited;  the  limit  corresponds  to  a  degree  of  com- 
bination higher  as  the  temperature  increases. 

Between  the  temperatures  r  and  0  the  compound  is  indestructible; 
the  elements  combine;  this  reaction  stops  only  when  the  combination 
is  complete. 

1  ARMAND  GAUTIER  and  H.  HELIER,  Comptes  Rendus,  v.  122,  p.  566,  18^6; 
H.  HELIER,  Annales  de  Chimie  et  de  Physique,  7th  S.,  v.  10,  p.  521,  1897. 


GENUINE  FALSE  EQUILIBRIA.  387 

Above  the  temperature  6  the  compound  decomposes;  the  elements 
combine;  these  lioo  reactions  are  limited;  at  a  given  temperature 
the  same  state  of  equilibrium  limits  the  states  of  the  system  within 
which  a  decomposition  is  produced  and  the  states  within  which  a 
combination  is  produced;  this  state  of  equilibrium  corresponds  to  a 
decomposition  the  more  complete  as  the  temperature  is  higher;  tem- 
peratures above  6  form  properly  the  domain  of  DISSOCIATION. 

Let  us  take  water  as  an  example. 

It  has  been  known  since  Lavoisier's  time  that  at  low  tempera- 
tures water  does  not  decompose,  that  oxygen  and  hydrogen  do  not 
combine;  it  is  known  also  that  at  sufficiently  high  temperatures 
where  water  is  decomposable  oxygen  and  hydrogen  combine  in 
totality.  It  may  therefore  be  said  that  since  the  origin  of  chem- 
istry the  existence  of  temperatures  less  than  t  has  been  recognized, 
as  well  as  temperatures  between  r  and  0. 

In  demonstrating  that  water  was  dissociable  at  a  very  high 
temperature  (Arts.  49  and  50),  H.  Sainte-Claire  Deville  showed 
the  existence  of  temperatures,  above  0,  where  veritable  equilibria 
are  established. 

Finally,  A.  Gautier  and  Helier  have  explored  recently  the  zone 
of  limited  combination  comprised  between  t  and  T. 

At  atmospheric  pressure,  heat  a  mixture  containing  16  grammes 
of  oxygen  to  1  gramme  of  hydrogen. 

At  180°  the  oxygen  and  hydrogen  begin  to  combine;  at  200° 
the  combination  becomes  measurable;  by  employing  an  artifice 
of  which  we  shall  say  a  word  in  the  last  chapter  (Art.  320),  Gautier 
and  Helier  have  been  able  to  follow  the  phenomena  up  to  825° 
without  obtaining  explosion.  In  all  this  temperature  interval  the 
combination  of  hydrogen  and  oxygen  is  limited;  the  value  of  the 
ratio  x  of  the  mass  of  water  formed  to  the  possible  mass  of  water, 
which  limits  the  combination,  increases  with  the  temperature  as 
indicated  by  the  following  table. 

At  these  temperatures  the  water  vapor,  either  alone  or  mixed 
with  a  certain  quantity  of  explosive  gas,  is  undecomposable ;  the 
combination  of  oxygen  and  hydrogen  is  therefore  not  limited  by 
the  inverse  reaction,  but  by  a  region  of  false  equilibria.  The 
temperature  interval  within  which  were  made  Gautier  and  Heller's 
experiments  is  wholly  below  the  temperature  we  have  called  T; 


388 


THERMODYNAMICS  AND  CHEMISTRY. 


Tempera- 
tures. 

X 

Tempera- 
tures. 

X 

180°C. 

0.0004 

416°  C. 

0.3570 

200 

0.0012 

433 

0.3981 

239 

0.0130 

498 

0.5638 

260 

0.0160 

620 

0.8452 

331 

0.0978 

637 

0.8565 

376 

0.02511 

875 

0.9610 

the  temperature  r  is  therefore  higher  than  875°;  still  higher  on 
the  temperature  scale,  and  probably  above  1000°,  is  located  the 
temperature  we  have  called  6,  at  which  is  entered  the  region  of 
the  dissociation  of  water  vapor,  object  of  the  researches  of  H. 
Sainte-Claire  Deville. 

From  Helier's  researches,  the  value  of  x  which,  at  a  given 
temperature  less  than  r,  limits  the  combination  of  hydrogen  with 
oxygen  at  atmospheric  pressure,  changes  when  the  mixture,  instead 
of  containing  16  grammes  of  oxygen  to  2  grammes  of  hydrogen, 
contains  an  excess  of  one  of  these  component  gases ;  it  changes  also 
if  an  inert  gas,  as  nitrogen,  for  example,  is  added  to  the  mixture. 

289.  Action  of  oxygen  on  carbon  dioxide. — What  we  have 
just  said  on  the  subject  of  the  formation  of  water  vapor  and  its 
dissociation  may  be  almost  textually  repeated  for  what  concerns 
carbonic  anhydride. 

At  temperatures  below  a  certain  limit  t  the  carbonic  gas  is 
undecomposable,  the  oxide  of  carbon  does  not  combine  with 
oxygen;  at  temperatures  high  enough  to  be  included  between 
two  certain  limits  r  and  6  the  carbonic  gas  is  undecomposable, 
but  combines  integrally  with  oxygen;  these  facts  have  long  been 
known.  The  observations  of  H.  Sainte-Deville  Claire  have  shown 
us  (Aft.  51)  that  beyond  the  temperature  6  the  carbonic  gas  may 
dissociate;  veritable  equilibria  are  then  established.  Helier  has 
made  known  the  region  of  limited  combination,  comprised  be- 
tween t  and  r. 

He  has  found  that  when  a  m'xture  containing  two  molecules 
of  oxide  of  carbon  and  one  molecule  of  oxygen  are  heated  with 
sufficient  precautions  to  avoid  any  explosion,  the  formation  of 
carbonic  acid  ceased  when  the  ratio  x  of  the  mass  of  this  acid 
formed  to  the  possible  mass  had  attained  a  certain  value,  variable 


GENUINE  FALSE  EQUILIBRIA. 


389 


with  the  temperature  and  increasing  with  it,  as  the  following 
table  shows: 


Tempera- 
tures. 

X 

Tempera- 
tures. 

X 

195°  C. 

0.0013 

504°  C. 

0.0730 

302 

0.0044 

566 

0.1443 

365 

0.0101 

575 

0.1727 

408 

0.0303 

600 

0.2114 

418 

0.0341 

689 

0.4636 

468 

0.0464 

788 

0.6030 

500 

0.0620 

855 

0.6500 

Throughout  this  temperature  interval  the  carbonic  gas  is 
undecomposable,  so  that  the  formation  of  this  substance  is  limited 
not  by  the  inverse  action,  but  by  the  establishment  of  a  false  equi- 
librium. It  is  only  at  very  much  higher  temperatures  that  we 
penetrate,  as  was  demonstrated  by  H.  Sainte-Claire  Deville,  into 
the  region  of  dissociation  for  carbonic  acid  gas. 

290.  Analogous  phenomena  shown  by  endothermic  combina- 
tions.— A  compound  formed  from  its  elements  with  absorption  of 
heat  may  very  well  present  phenomena  analogous  to  those  we  have 
just  described  for  an  exothermic  compound;  for  Fig.  121  we  should 
then  substitute  a  representation  such  as  Fig.  122.  Here  also  x  con- 


Decomposition 


Equili 


'Combination. 


FIG.  122. 

tinues  to  denote  the  ratio  of  the  mass  of  the  compound  existing 
in  the  system  to  the  possible  mass. 

There  are  four  regions  to  distinguish : 

1°.  At  temperatures  below  t  no  reaction  is  produced  in  a  system 
containing  the  compound  and  the  elements  capable  of  forming  ii 
whatever  the  value  of  x; 


390  THERMODYNAMICS  AND  CHEMISTRY. 

2°.  At  temperatures  included  between  t  and  r  the  compound  cannot 
be  formed  at  the  expense  of  its  elements;  on  the  contrary,  if  the  initial 
value  of  x  is  sufficiently  great,  the  compound  is  partly  destroyed;  the 
composition  is  limited;  the  value  of  x  which  limits  the  decomposition 
is  the  less  as  the  temperature  is  higher;  the  reaction  is  limited  not  by 
the  inverse  action,  but  by  the  production  of  false  equilibria; 

3°.  At  temperatures  between  r  and  6  the  compound  cannot  be 
formed;  it  decomposes;  this  reaction  is  unlimited; 

4°.  At  temperatures  above  0,  according  to  the  value  possessed  by 
the  ratio  x  in  the  system,  the  latter  may  be  the  seat  either  of  a  decom- 
position or  of  a  combination;  at  a  given  temperature  the  two 
reactions,  the  inverse  of  each  other,  are  limited  by  the  same  value  of. 
x;  this  limiting  value  of  x  increases  with  the  temperature. 

These  distinctions  enable  us  to  classify  the  properties  of  a  great 
number  of  endothermic  compounds. 

291.  Ozone. — For  ozonized  oxygen    ordinary  temperature  is 
already  higher  than  the  temperature  T;  at  this  temperature,  more 
rapidly  at  100°,  and  still  more  so  at  200°,  ozone  undergoes  decom- 
position which  may  be  regarded  as  complete;   we  know,  further, 
that  Troost  and  Hautefeuille  (Art.  176)  have   shown   the  direct, 
but  partial,  transformation  of  oxygen  into  ozone  at  temperatures 
of  about  1200°;  these  temperatures  are  therefore  higher  than  6. 

292.  Silicon  trichloride.     Investigations  of  Troost  and  Haute- 
feuille.— Troost  and  Hautefeuille  were  able  to  completely  explore 
the  various  parts  of  the  field  represented  by  Fig.  122  for  certain 
silicon  compounds,1  and  in  particular  for  silicon  trichloride,  Si2Cl6. 

At  250°  this  substance  is  not  formed  by  the  action  of  tetrachlo- 
ride  vapor,  SiCl4,  on  silicon;  in  return,  silicon  trichloride  vapors 
are  undecomposable  at  this  temperature. 

At  350°  the  trichloride  vapor  undergoes  decomposition  very 
slowly  and  the  reaction  is  limited;  the  deposit  of  silicon  on  the 
walls  of  the  vase  is  hardly  sensible  after  24  hours ;  the  tetrachloride 
vapors  are  still  without  action  on  silicon. 

When  the  temperature  is  raised,  the  decomposition  of  the  tri- 
chloride of  silicon  into  the  tetrachloride  and  silicon  becomes  more 


1  TROOST  and  HAUTEFEUILLE,  Annales  de  Chimie  et  de  Physique,  5th  S., 
v.  7,  1876. 


GENUINE  FALSE  EQUILIBRIA.  391 

and  more  marked ;  it  destroys  at  440°  a  notable  fraction  of  the  com- 
pound substance ;  at  800°,  if  the  experiment  is  sufficiently  prolonged, 
the  decomposition  is  complete. 

On  the  contrary,  at  1000°  the  decomposition  is  but  partial;  on 
the  other  hand,  the  tetrachloride  combines  at  this  temperature 
partially  with  silicon,  giving  the  trichloride. 

293.  Systems  with   unlimited   reaction  and  the  principle  of 
maximum  work. — A  considerable  number  of  chemical  reactions 
are  classed  in  the  category  of  which  the  formation  of  sulphuretted 
hydrogen  and  the  decomposition  of  silicon  trichloride  are  types. 
All  these  reactions  give  rise  to  an  important  observation:    At  tem- 
peratures included  between  t  and  T,  when  the  only  possible  reaction  is 
limited  by  states  of  false  equilibrium,  and  at  temperatures  between  r  and 
6.  where  this  reaction  is  unlimited,  it  is  exothermic,  so  that  the  principle 
of  maximum  work  is  verified;  to  find  the  principle  of  maximum  work 
in  default  it  is  necessary  to  attain  temperatures,  above  0,  where  states 
of  veritable  equilibrium  may  be  established. 

294.  Systems  with  unlimited  reaction  are  not  essentially  dis- 
tinct from  systems  with  limited  reaction. — It  seems  at  first  sight 
that  a  radical  difference  separates  the  systems  incapable  of  un- 
limited reaction  which  we  have  studied  in  Arts.  285  and  286,  from 
those  which    may,  between    certain    temperatures,  give    rise    to 
unlimited  reactions;  such  are  the  systems  studied  in  Arts.  287  to 
293.     In  reality,  as  we  shall  show,  it  may  be  very  well  admitted  that 
such  a  difference  is  a  difference  not  hi  nature,  but  in  degree. 

Take  a  system  which  contains  nearly  perfect  gases  and  where  an 
exothermic  compound  may  be  produced  or  dissociated ;  for  instance, 
a  system  containing  sulphur  in  liquid  and  vapor  states,  hydrogen 
and  hydrogen  sulphide ;  let  x  be  the  ratio  between  the  mass  of  the 
compound  the  system  contains  and  the  mass  of  this  same  com- 
pound it  would  contain  if  the  combination  of  its  elements  were  pushed 
as  far  as  possible;  heat  the  system  either  at  constant  pressure  or 
at  constant  volume.  The  line  W  (Fig.  123)  of  veritable  equilibria 
has,  in  the  plane  TOx,  a  form  we  have  already  traced  in  Fig.  Ill; 
as  far  as  point  B,  of  abscissa  d,  it  remains  practically  identical  with 
the  line  A  A' ,  parallel  to  OT,  and  having  as  constant  ordinate  x  =  l ; 
it  then  detaches  itself  and  descends  from  left  to  right. 

Let  CC'  be  the  line  separating  the  region  of  false  equilibria  from 


3S2 


THERMODYNAMICS  AND  CHEMISTRY. 


the  region  of  combination,  and  DD'  the  line  which  separates  the 
region  of  false  equilibria  from  that  of  decomposition ;  if  things  hap- 


A' 


e  c 


FIG.  123. 


pen  in  accordance  with  what  we  have  seen  in  Art.  286,  these  two 
lines  should  coincide  with  the  line  of  veritable  equilibria  VV  beyond 
a  certain  point  P  of  abscissa  T. 

The  temperature  r  may  be  considerably  above  6 ;  we  obtain  then 
an  arrangement  analogous  to  that  we  studied  in  the  system  hydro- 
gen, selenium,  selenhydric  acid;  the  new  arrangement  is  simply 
symmetrical  with  the  other  with  respect  to  an  axis  parallel  to  OT. 


Decomposition^ 


FIG.  124. 


The  temperature  r  may,  on  the  contrary,  be  considerably  below 
0,  the  point  P  being  well  to  the  left  of  the  point  B;  in  this  case, 
shown  in  Fig.  124,  the  line  CC'  alone  is  discernible;  the  line  DD'  is 
reduced  to  a  negligible  segment  DP.  Practically  the  chemical 
statics  of  our  system  is  resumed  in  the  propositions  stated  at  the 
close  of  Art.  287. 


GENUINE  FALSE  EQUILIBRIA.  393 

295.  One  may  always  cool  a  chemical  system  sufficiently  for 
it  to  exist  in  the  state  of  false  equilibrium. — We  have  therefore 
only  three  kinds  of  temperature  to  distinguish  in  the  study  of  a 
chemical  system: 

1°.  High  temperatures,  where  the  system  is  susceptible  of  two 
reactions,  the  inverse  of  each  other,  having  as  common  limit  a  series 
of  states  of  veritable  equilibrium; 

2°.  Moderate  temperatures,  where  the  system  can  undergo  two 
reactions,  limited  no  longer  by  the  inverse  reaction  but  by  states 
of  false  equilibria. 

3°.  Low  temperatures,  where  the  system  is  not  susceptible  of 
any  reaction. 

All  these  observations  seem  to  accord  with  this  principle: 
Given  a  chemical  system,  the  temperature  may  always  be  sufficiently 
lowered  to  cause  the  system  to  remain  in  a  state  of  false  equilibrium. 

Thus  below  250°  a  mixture  containing  only  selenium  and 
hydrogen,  without  trace  of  selenhydric  acid,  remains  in  a  state  of 
false  equilibrium;  no  reaction  is  produced  in  it;  below  215°  a 
system  containing  sulphur  and  hydrogen  is  in  the  state  of  false 
chemical  equilibrium. 

296.  False    equilibria    at   very   low  temperatures.     Pictet's 
researches. — For    certain    systems    the   state    of   false    chemical 
equilibrium  cannot  be  obtained  except  by  greatly  lowering  the  tem- 
perature;  this  was  shown  by  R.  Pictet.1     At  —125°  a  mixture  of 
frozen  sulphuric  acid  and  caustic  soda  may  be  compressed  without 
any  reaction  being  produced;  as  long  as  the  temperature  is  below 
—  80°  C.  no  combination  takes  place;   it  is  produced  abruptly  at 
this  temperature  of  —80°  C.,  liberating  such  a  quantity  of  heat  that 
the  eprouvette  containing  the  mixture  is  broken. 

Sulphuric  acid  and  potash  remain  in  equilibrium  at  tempera- 
tures less  than  —  90°  C. ;  sulphuric  acid  and  a  concentrated  ammo- 
niacal  solution  at  temperatures  less  than  —65°  C.;  at  —120°  C. 
sulphuric  and  hydrochloric  acids  leave  litmus  its  blue  color;  lit- 
mus turns  red  at  —  110°C.  with  hydrochloric  acid,  and  at  — 105°  C. 
with  sulphuric  acid. 

It  is  to  be  remarked  that  certain  of  the  systems  of  which  we 
have  just  spoken  are  not,  perhaps,  at  the  temperatures  realized 
1  R.  PICTET,  Comptes  Rendus,  v.  115,  p.  814,  1892. 


394  THERMODYNAMICS  AND  CHEMISTRY. 

by  Pictet,  systems  in  equilibrium,  but  merely  systems  where  an 
excessively  slow  reaction  is  produced ;  according  to  Besson J 
hydrochloric  acid  which  has  remained  at  a  very  low  temperature, 
at  —80°  for  instance,  in  contact  with  sodium,  encloses  small 
quantities  of  sodium  chloride. 

297.  The  reaction-point. — Take,  at  a  very  low  temperature, 
a  system  in  the  state  of  false  equilibrium  and  gradually  raise  the 
temperature;  at  a  certain  moment  the  system  will  cease  to  be 
in  false  equilibrium  and  a  reaction  will  be  produced.  The  tempera- 
ture at  which  a  given  system,  under  a  given  pressure  or  main- 
tained at  a  given  volume,  ceases  to  be  in  the  state  of  false  equi- 
librium and  becomes  the  seat  of  a  chemical  modification,  is  called 
the  reaction-point  of  this  system.  Thus  the  reaction-point  of  a 
system  which  contains  hydrogen  and  selenium,  without  trace  of 
selenhydric  acid,  and  which  is  heated  at  constant  volume,  is  close 
to  250°;  at  this  temperature  selenhydric  acid  begins  to  be  formed. 

For  certain  systems  the  reaction-point  may  correspond  to  a 
very  low  temperature;  we  have  seen  that  the  litmus  reaction  with 
-hydrochloric  acid  was  about  —110°  C. 

In  other  cases,  on  the  contrary,  this  reaction-point  corresponds 
to  an  extremely  elevated  temperature;  one  of  these  cases  is  given 
us  by  a  mixture  of  hydrogen  and  nitrogen. 

Ammonia  gas  would  be  formed,  starting  with  hydrogen  and 
nitrogen,  with  a  great  liberation  of  heat;  if,  therefore,  a  mixture 
of  these  three  gases  kept,  either  at  constant  pressure  or  at  constant 
volume,  was  in  a  state  of  veritable  equilibrium,  the  combination 
in  it  would  be  almost  complete  at  low  temperature;  it  is  only  at 
a  high  temperature  that  ammonia  gas  would  show  appreciable 
dissociation. 

In  fact  a  mixture  of  hydrogen  and  nitrogen,  whether  or  not 
containing  ammonia  gas,  may  be  kept  in  the  state  of  false  equilib- 
rium at  almost  any  of  the  temperatures  produced  by  our  furnaces  ; 
it  is  only  at  very  high  temperatures,  generated  by  very  hot  electric 
sparks,  that  the  combination  begins  to  take  place,  as  was  shown 
by  Morren ; 2  his  observation  was  confirmed  by  means  of  the  hot 
and  cold  tube  apparatus  of  H.  Sainte-Claire  Deville.3 

1  BESSON,  Comptes  Rendus,  v.  124,  p.  763,  1897. 
*  MORREN,  Comptes  Rendus,  v.  48,  p.  342,  1859. 
9  H.  SAINTE-CLAIRE  DEVILLE,  Lefons  sur  la  dissociation,  1864. 


GENUINE  FALSE  EQUILIBRIA. 


395 


The  reaction-point  of  a  system  may  depend  upon  a  number 
of  circumstances :  on  the  initial  pressure  supported  by  the  system 
if  heated  at  constant  volume ;  on  the  volume  if  heated  at  constant 
pressure;  on  the  initial  composition  of  the  system  and  the  foreign 
substances  which  may  be  mixed  with  it. 

Finally,  in  certain  cases  these  various  circumstances  may 
influence  not  only  the  temperature  of  the  reaction,  but  also  the 
nature  of  the  reaction  which  begins  to  be  produced  the  instant 
this  temperature  is  reached. 

Take,  for  example,  a  system  formed  of  hydrogen,  selenium, 
and  selenhydric  acid.  Trace  (Fig.  125)  the  line  CC',  which  sepa- 


Region  of 
Decomposition 


rates  the  region  of  combination  from  the  region  of  false  equilibria, 
and  the  line  DDf,  which  separates  the  region  of  decomposition 
from  the  region  of  false  equilibria;  this  last  line  has  a  point  M 
lower  than  all  the  others;  let  /*=0/*  be  the  ordinate  of  this  point. 

Consider  a  systen  in  which  the  initial  value  r  of  the  ratio*/?  is 
less  than  /£;  suppose  the  temperature  low  enough  for  the  system 
to  be  in  the  state  of  false  equilibrium  and  gradually  raise  this 
temperature;  the  representative  point  describes  the  straight  line 
rf  which  meets  the  line  CC'  at  7-;  the  abscissa  t  of  the  point  7-  is 
the  reaction-point  of  the  system;  at  the  instant  the  temperature 
reaches,  then  exceeds  this  value  t,  the  system  becomes  the  seat 
of  a  combination. 

Take,  on  the  contrary,  a  system  in  which  the  initial  value  of 
R  of  the  ratio  p  is  greater  than  /*  and  whose  temperature  is  low 


396  THERMODYNAMICS  AND  CHEMISTRY. 

enough  for  equilibrium  to  be  reached;  when  the  temperature  in- 
creases, the  representative  point  will  describe  the  line  Rd,  parallel 
to  OT,  meeting  DD'  in  d;  the  abscissa  T  of  the  point  d  will  be  the 
reaction-point  of  the  system;  the  system,  when  it  attains  this 
point,  becomes  the  seat  of  a  decomposition. 

298.  Reaction-point  for  the  phosphorescence  of  phosphorus. 
Joubert's  studies. — For  the  majority  of  cases  the  complexity  is 
less;  at  the  moment  when  the  syst  m  reaches  the  reaction-point 
there  is  produced  a  reaction  whose  nature  does  not  depend  upon 
the  initial  composition  of  the  system. 

Thus,  whatever  the  initial  composition  of  a  system  containing 
sulphur,  hydrogen,  sulphuretted  hydrogen,  the  reaction-point 
always  corresponds  to  the  beginning  of  combination. 

The  same  is  true  in  the  case  of  the  combination  of  oxygen  and 
phosphorus,  studied  in  detail  by  Joubert.1 

Consider  a  space  which  contains  oxygen  and  the  saturated 
vapor  of  phosphorus,  in  the  presence  of  an  excess  of  phosphorus. 
The  oxygen  and  phosphorus  may  combine  either  rapidly,  which 
constitutes  the  phenomenon  of  the  combustion  of  phosphorus,  or 
slowly,  which  produces  phsophorescence. 

In  such  a  system  there  exists  a  reaction-point;  below  this 
temperature  no  combination  is  produced  in  the  system;  above, 
phosphorescence  is  produced  and  then  combustion. 

This  reaction-point  is  not  fixed;  it  depends  upon  the  pressure 
supported  by  the  system;  it  is  the  higher  as  the  pressure  is  higher. 

Take  as  abscissae  the  pressures  n  (Fig.  126),  as  coordinates  the 
temperatures  T.  For  every  pressure  it  there  corresponds  a  reaction- 
point  T]  the  point  M,  of  coordinates  x,  T,has  a  certain  curve  CC' 
for  locus. 

This  curve  divides  the  plane  into  two  regions. 

Take  a  point  a  of  abscissa  TT,  of  ordinate  0,  less  than  T,  reac- 
tion-point at  the  pressure  TT;  this  point  represents  a  system  in 
which  no  reaction  is  produced ;  the  point  a  is  therefore  in  the  region 
of  false  equilibria,  which  coincides  with  the  region  situated  below 
the  curve  CC'. 

Take,  on  the  contrary,  a  point  A,  of  abscissa  TT  and  ordinate  0, 

1  JOUBERT,  Annales  de  I'Ecole  normale  superieure,  2d  Series,  v  3,  p.  209, 
1874. 


GENUINE  FALSE  EQUILIBRIA. 


397 


greater  than  T;  this  point  represents  a  system  in  which  oxygen 
and  phosphorus  combine;  the  point  A  is  therefore  in  the  region 
of  combination,  which  coincides  with  the  region  located  above  the 
curve  CC'. 


Combination 


II 


n 


FIG.  126. 

The  curve  CC'  rises  from  left  to  right;  the  region  of  combination 
is  therefore  to  the  left  of  the  curve  CC',  and  the  region  of  false  equilibria 
to  the  right  of  the  same  curve. 

Whence,  if  we  take  a  point  b,  of  ordinate  T  and  of  abscissa  p, 
less  than  n,  this  point  represents  a  system  where  the  oxygen  and 
the  phosphorus  combine;  a  point  B,  of  the  same  ordinate  T, 
but  of  abscissa  P,  greater  than  TT,  represents  a  system  where  no 
reaction  takes  place.  Therefore,  at  every  temperature  T  corre- 
sponds a  certain  limiting  pressure  it;  under  a  pressure  less  than  n 
oxygen  combines  with  phosphorus;  at  a  pressure  greater  than  K  a, 
system  containing  oxygen  and  phosphorus  is  in  equilibrium;  the 
pressure  K  is  the  higher  as  the  temperature  is  higher. 

This  is  the  law  stated  and  verified  by  Joubert,  who  gives  the 
following  values  for  TT: 


Tempera- 
tures. 

1C 

Tempera- 
tures. 

1C 

1°.4C. 

355  mm. 

9°.3C. 

538mm. 

3  .0 

387 

11  .5 

580     " 

4  .4 

408 

14  .2 

650     " 

5  .0 

428 

18  .0 

730     " 

6  .0 

460 

19  .2 

760     " 

8  .9 

519 

398  THERMODYNAMICS  AND  CHEMISTRY. 

The  line  CC"  is  sensibly  straight. 

The  form  and  position  of  this  line  vary  greatly  when  certain 
inert  gases  are  mixed  with  the  oxygen.  Joubert  has  made  a  very 
complete  study  of  this  variation. 

The  combustion  of  phosphoretted  hydrogen  in  oxygen  has 
given  rise  to  analogous  observations  on  the  part  of  Van  de  Stadt.1 

299.  Analogy  of  the  states  of  false  equilibria  with  the  mechan- 
ical equilibria  due  to  friction. — The  several  examples  we  have  just 
studied  give  us  a  clear  idea  of  the  principal  characteristics  of  false 
equilibria;  in  particular  they  show  us  readily  that,  in  a  system 
capable  of  false  equilibria,  the  condition  for  equilibrium  is  not 
expressed  by  an  equality,  but  by  an  inequality  or  by  a  double 
inequality. 

Is  this  characteristic  incompatible  with  the  analogy  between 
chemical  statics  and  statics  properly  so  called,  analogy  which  we 
regard  as  one  of  the  guiding  ideas  of  the  science?  Quite  the  contrary, 
and  this  characteristic  establishes  a  close  resemblance  between  the 
chemical  systems  capable  of  false  equilibria  and  the  mechanical 
systems  possessing  friction. 

Let  us  take  the  following  example,  ingeniously  imagined  by 
Pelabon: 

Consider  a  cylinder  full  of  air  with  axis  vertical,  closed  at  the 
base ;  within  this  cylinder  moves  a  piston  upon  which  weights  may 
be  put;  to  simplify,  suppose  the  area  of  the  cross-section  of  the  cyl- 
inder to  be  unity. 

Denote  by  H  the  pressure  of  the  gaseous  atmosphere  above  the 
piston,  by  (u  the  weight  of  the  latter,  by  p  the  additional  weight  it 
carries. 

If  the  friction  on  the  walls  of  the  tube  is  neglected,  there  is  for 
each  weight  p  one  equilibrium  position  for  the  piston  and  only  one; 
for  example,  if  V  represent  the  distance  from  the  base  of  the  piston 
to  the  bottom  of  the  cylinder  when  p=0,  and  x  the  value  of  the 
same  length  when  the  additional  weight  has  the  value  p,  we  shall 
have,  applying  Boyle's  law  to  the  gaseous  mass  enclosed  in  the 
cylinder,  the  relation 

(3)  (H+oj)V=(H+aj+p)x, 

which  will  determine  the  position  of  equilibrium  in  question. 
1  VAN  DE  STADT,  Zeitschrift  fur  physikalische  Chemie,  v.  12,  p.  322,  1893. 


GENUINE  FALSE  EQUILIBRIA. 


399 


Piston_de8eendB 


*' 


Take  two  rectangular  coordinate  axes  (Fig.  127);  along  the 
abscissa  axis  Op  lay  off  the  values  of  the  additional  weight  p;  as 
ordinate  take  the  corresponding 
values  of  x  calculated  by  equation 
(3) ;  the  point  v,  of  coordinates  p, 
x  will  represent  an  equilibrium 
state  of  the  piston  supposed  with- 
out friction;  as  p  varies,  the  point 
v  describes  a  line  Wl}  which  we 
shall  call  the  line  of  veritable 
equilibrium  of  the  piston. 

Suppose  now  the  piston  rubs 
along  the  interior  of  the  cylinder,  Fl°-  127- 

and  denote  by  P  the  additional  weight;  hi  order  for  equilibrium 
to  exist  it  will  be  no  longer  necessary  for  the  pressure  (H+w+P) 

V 

exerted  by  the  cylinder  to  be  equal  to  the  pressure  (H+co)—  of  the 

gaseous  mass;  it  will  be  sufficient  for  the  absolute  value  of  the 
difference  of  these  two  pressures  not  to  exceed  a  magnitude  <£, 
dependent  upon  the  nature  of  the  contiguous  surfaces  of  the 
cylinder  and  piston.  The  condition  for  equilibrium  of  the  piston  is 
then,  taking  friction  into  account, 


(4) 


This  may  be  written  otherwise.    Denote  by 


(5) 


the  value  of  P  which  would  correspond  to  the  state  of  veritable 
equilibrium  for  which  the  bottom  of  the  cylinder  and  base  of  the 
piston  are  at  a  distance  x;  the  point  v,  of  coordinates  (p,  x),  will  be 
evidently  the  point  of  ordinate  x  on  the  line  of  veritable  equilibria, 
for  equation  (5)  is  but  equation  (4)  solved  for  p.  The  double  in- 
equality (4)  may  be  written 


(6) 


400  THERMODYNAMICS  AND  CHEMISTRY. 

Through  the  point  v  draw  a  parallel  to  the  axis  OP;  on  this  line 
mark  off  two  points  ra,  d  having  respectively  as  abscissae  (p—<f>) 
and  (p-f-0);  the  distances  dv,  vm  are  both  equal  to  </>;  every  point 
of  the  segment  md  represents  a  state  of  equilibrium  of  the  piston. 

A  point  on  the  line  xm,  situated  to  the  left  of  the  point  m, 
represents  a  system  where  the  distance  from  the  base  of  the  piston 

to  the  bottom  of  the  cylinder  is  x,  but  where  the  pressure  of  the  gas 
y 

(H+aj)—  exceeds  the  pressure  (H+P+aj)  exerted  by  the  piston 
x 

by  a  quantity  greater  than  <f> ;  in  these  conditions  the  gas  expands 
and  the  piston  rises. 

A  point  on  the  line  dx'}  situated  to  the  right  of  the  point  d, 
represents  a  system  where  the  distance  from  the  base  of  the  piston 
to  the  bottom  of  the  cylinder  is  still  x,  but  where  the  pressure 
>)  exerted  by  the  piston  exceeds  the  pressure  of  the  gas 
y 
—  by  a  quantity  greater  than  <£;  in  these  conditions  the  gas 

is  compressed  and  the  piston  descends. 

For  each  value  of  x  analogous  considerations  may  be  repeated. 

When  x  is  varied,  the  point  m  describes  a  line  Mm,  and  the  point 
d  a  line  Dd;  these  two  lines  divide  the  plane  into  three  regions; 
every  point  of  the  region  situated  to  the  left  of  the  line  Mm  repre- 
sents a  system  where,  without  initial  speed,  the  piston  rises;  every 
point  of  the  region  situated  to  the  right  of  the  line  Dd  represents  a 
system  where,  without  initial  speed,  the  piston  descends;  finally, 
every  point  in  the  region  situated  between  these  two  lines,  including 
the  points  on  the  lines,  represents  a  system  where,  without  initial 
speed,  the  piston  rests  stationary;  this  is  the  region  of  false  equilib- 
ria. 

The  line  of  veritable  equilibria  is  in  its  entirety  drawn  in  the 
region  of  false  equilibria. 

This  example  renders  tangible  the  analogy  which  exists  between 
mechanical  systems  with  friction  and  chemical  systems  with  false 
equilibria. 

300.  The  existence  of  false  equilibria  in  chemical  systems  is 
not  exceptional,  but  regular. — The  existence  of  friction  in  a  mech- 
anism must  not  be  regarded  as  an  exception,  but  the  rule ;  in  a  great 
number  of  cases  this  friction  is  feeble  enough  to  be  neglected, 


GENUINE  FALSE  EQUILIBRIA. 


401 


and  freed  from  this  complication  the  laws  of  mechanics  assume 
the  simple  form  in  which  they  are  ordinarily  exposed;  but  it 
would  be  dangerous  to  forget  that  these  forms  are  incomplete  and 
constitute,  in  the  most  favorable  cases,  but  an  approximation; 
one  would  be  led  otherwise  to  seek  for  a  perpetual  motion. 

Analogy  leads  us  to  suppose  that  false  chemical  equilibria 
are  not  exceptional  facts,  but  are  the  rule;  every  chemical  system 
is  capable  of  possessing  such  states  of  equilibrium ;  only,  in  a  great 
number  of  cases  the  states  of  false  equilibrium  are  all  so  near  to 
the  state  of  veritable  equilibrium  that  one  may  not  practically 
distinguish  them  from  this  last,  which  alone  seems  realizable. 

Thus  Pelabon's  experiments  do  not  give  evidence  of  a  region 
of  false  equilibrium,  within  the  system  formed  of  hydrogen,  sele- 
nium, and  selenhydric  acid,  at  temperatures  above  325°.  But 
these  results  may  very  well  be  interpreted  in  the  following  way: 
The  two  curves  CC',  DD'  (Fig.  128)  which  limit  the  region  of 


FIG.  128. 

false  equilibria  remain,  at  every  temperature,  distinct  from  each 
other  and  from  the  line  VV  of  veritable  equilibrium;  but  at  tem- 
peratures above  325°  the  two  lines  CC',  DD'  are  too  close  together 
and  the  region  of  false  equilibria  is  reduced  to  a  too  narrow  band 
to  permit  distinguishing  by  experiment  the  false  from  the  veri- 
table equilibria. 

This  manner  of  looking  at  the  matter  carries  with  it  a  pro- 
found change  in  the  ideas  we  have  admitted  so  far  concerning 
chemical  equilibrium.  The  state  of  chemical  equilibrium  has 


402  THERMODYNAMICS  AND  CHEMISTRY. 

appeared  (Chap.  IV,  Art.  61)  as  the  common  frontier  between 
the  states  where  the  system  undergoes  a  modification  in  a  definite 
direction,  and  the  states  where  the  system  undergoes  a  modifica- 
tion in  the  opposite  direction;  it  was  the  state  into  which  two 
reactions  of  opposite  dir  ctions  limit  each  other;  its'  essential  prop- 
erty is  expressed  by  this  proposition :  A  series  of  equilibrium  states 
is  a  reversible  transformation. 

These  ideas  grouped  about  the  notion  of  reversibility  appear 
to  us  now  as  notions  incapable  of  representing  exactly  the  facts; 
chemical  statics  constructed  by  means  of  these  notions  is  a  too 
simple  statics;  it  gives  only  laws  for  an  ideal  case,  for  a  limiting 
case  to  which  certain  systems  approach  more  or  less.  Similarly, 
mechanics  where  abstraction  is  made  of  friction  is  a  mechanics 
too  simplified ;  its  laws  are  limiting  laws,  to  which,  in  certain  cases, 
the  real  laws  of  motion  approach  more  or  less. 

A  last  similarity  is  to  be  noticed  between  the  evolution  of  me- 
chanics and  the  development  of  chemical  mechanics. 

The  mechanical  systems  which  ordinarily  surround  us  are 
rendered  extremely  complex  by  the  continual  presence  of  friction; 
thus,  unless  one  attacks,  with  Kepler  and  Galileo,  celestial 
mechanics  free  from  friction,  one  is  compelled  to  consider  groups 
of  bodies  susceptible  of  rubbing  against  each  other  and  it  becomes 
impossible  to  discover  simple  laws,  such  as  the  law  of  inertia 
which  must  serve  as  the  basis  for  dynamics — for  a  dynamics  no 
doubt  too  abstract  and  ideal,  but  whose  creation  had  necessarily  to 
precede  the  theory  of  friction. 

Similarly,  chemical  actions  which  are  produced  at  ordinary 
temperatures  give  rise  to  continued  false  equilibria.  As  long  as 
they  alone  were  considered,  chemical  mechanics  could  not  be 
placed  upon  a  secure  foundation.  The  principles  of  this  science 
were  not  clearly  appreciated  until  after  H.  Sainte-Claire  Deville, 
by  creating  the  chemistry  of  high  temperatures,  had  eliminated 
chemical  friction. 


CHAPTER  XIX. 


UNEQUALLY  HEATED  SPACES. 

301.  Formation  and  dissociation  of  selenhydric  acid  in  an 
unequally  heated  space.  Three  cases  to  distinguish. — The  prin- 
ciples we  have  just  been  stating  lend  themselves  to  the  discussion 
of  phenomena  produced  in  unequally  heated  spaces. 

We  shall  take  as  example  the  formation  of  selenhydric  acid 
from  selenium  and  hydrogen,  and  we  shall  neglect  the  volatility 
of  selenium;  this  volatility  affects  the  laws  we  are  going  to  state 
by  slight  perturbations  easy  to  allow  for;  but  in  neglecting  it  we 
shall  have  the  advantage  of  obtaining  expressions  applicable  to 
non-volatile  or  very  slightly  volatile  substances. 

Take  the  temperatures  T  as  abscissae,  and  as  ordinates  the  ratio  x 
of  the  mass  of  the  compound  existing^ 
in  the  system  to  the  mass  which  would 
be  found  there  if  the  combination  were 
pushed  as  far  as  possible  (Fig.  129). 

We  shall  suppose  the  selenium,  hy- 
drogen and  selenhydric  acid  enclosed  in 
a  tube  where  the  temperature  varies  be- 
tween a  lower  limit  T0  (temperature  of 
the  cold  extremity)  and  a  higher  limit  T 
(temperature  of  the  hot  extremity). 

Due  to  the  diffusion  of  the  gases,  the  ratio  x,  here  equal  to 


the  ratio  p 


considered  by  Pe"labon,  has  sensibly  the  same 


, 

value  in  all  the  parts  of  the  gaseous  mass  at  a  given  instant;  the 
various  parts  of  the  tube  correspond,  therefore,  at  the  same  instant, 
to  different  values  of  T,  but  to  the  same  value  of  x',  they  are  repre- 

403 


404  THERMODYNAMICS  AND  CHEMISTRY. 

sented  by  the  various  points  of  a  straight  line  AB,  parallel  to  OT, 
whose  extremities  A  and  B  have  respectively  for  abscissae  T0  and 

TV 

Suppose  any  portion  whatever  of  the  line  A  B  to  be  traced  in 
the  region  of  decomposition;  in  the  portions  of  the  heated  tube 
represented  by  the  points  of  the  straight  line  AB  situated  in  the 
region  of  decomposition,  selenhydric  acid  is  certainly  split  into 
selenium  and  hydrogen;  whence  this  first  proposition:  The  system 
contained  in  the  unequally  heated  tube  cannot  be  in  equilibrium  as 
long  as  some  point  of  the  representative  straight  line  AB  exists  in 
the  region  of  decomposition. 

Suppose,  in  the  second  place,  certain  points  of  the  repre- 
sentative straight  line  AB  are  in  the  region  of  combination;  if 
in  the  portions  of  the  tube  represented  by  these  points  there  is 
solid  or  liquid  selenium,  selenhydric  acid  will  be  formed  in  these 
portions:  the  system  contained  in  the  uneqally  heated  tube  cannot 
be  in  equilibrium,  therefore,  so  long  as  there  exists  solid  or  liquid 
selenium  in  a  portion  of  the  tube  represented  by  a  point  in  the  region 
of  combination. 

These  principles  stated,  let  us  examine  some  problems  which 
have  been  studied  experimentally  by  Pelabon.1 

Let  us  turn  back  to  the  sketch  (Fig.  130)  which  gives,  for  the 
systems  studied,  the  disposition  of  the  line  of  veritable  equilibria, 
of  the  region  of  false  equilibria,  of  the  region  of  decomposition,  and 
of  the  region  of  combination.  Several  temperatures  merit  remark. 

The  two  limiting  lines  of  the  false  equilibria  coalesce  practically 
with  the  line  of  veritable  equilibria  at  a  point  P  whose  abscissa  does 
not  exceed  350°. 

The  line  PV  of  the  veritable  equilibria  has  a  point  M,  of  maxi- 
mum ordinate,  whose  abscissa  is  close  to  575°. 

The  line  DP  has  a  point  m,  of  minimum  ordinate;  the  abscissa 
of  this  point  is  sensibly  270°;  if,  through  the  point  m,  the  tangent 
to  the  line  DP  is  drawn,  tangent  which  is  parallel  to  OT,  this  tangent 
cuts  the  line  CP  in  a  point  n  whose  abscissa  is  310°. 

Finally,  the  point  C  of  the  axis  OT,  where  the  line  CP  starts, 
corresponds  to  a  temperature  less  than  250°. 

1  H.  PELABON,  Sur  la  dissociation  de  I'acide  selenhydrique  (Me"m.  de  la 
Soc.  d.  Sciences  physiques  et  naturelles  de  Bordeaux,  5th  S.,  v.  3,  p.  232). 


UNEQUALLY  HEATED  SPACES. 


405 


The  experiments  made  by  Pelabon  are  concerned  with  the  three 
following  cases: 

FIRST  CASE. — All  portions  of  the  tube  are  at  temperatures  included 
between  350°  and  575°. 

The  straight  line  representing  the  state  of  the  system  in  equilib- 
rium cannot  in  any  way  encroach  on  the  region  of  decomposition; 
neither  can  it  be  entirely  traced  in  the  region  of  combination,  for 


.200     Q270310  350    400 

FIG.  130. 


500       575600 


700     T 


the  excess  of  liquid  selenium  would  be  found  in  a  portion  of  the 
tube  represented  by  a  point  in  the  region  of  combination;  therefore, 
in  order  to  have  equilibrium,  it  is  necessary  for  the  representative 
straight  line  AB  of  the  system  in  equilibrium  to  have  a  point  in 
common  with  the  line  of  veritable  equilibria  and  all  the  other  points 
in  the  region  of  combination ;  besides,  selenium  must  be  all  collected 
in  the  portion  of  the  tube  represented  by  the  first  point;  it  is  clear, 
furthermore,  that  this  disposition  assures  equilibrium. 

Whence  the  final  position  of  the  straight  line  AB  is  that  repre- 
sented by  Fig.  131 ;  at  the  end  of  the  experiment,  the  excess  of  selenium 
is  entirely  collected  in  the  coldest  region  of  the  heated  tube;  the  compo- 
sition of  the  system  is  the  same  as  if  it  had  been  maintained  wholly  at 
the  temperature  T0  of  the  cold  extremity;  we  may  say  that  the  prin- 
ciple of  Watt  or  of  the  cold  boundary  is  here  applicable. 

Pelabon's  observations  justifying  this  statement  are: 

!T0 =425°,     771  =  658°,    ,0=0.3678. 


406 


THERMODYNAMICS  AND   CHEMISTRY. 


A  tube  kept  at  the  same  time  entirely  at  T0=425°  would  give  for 
p  the  value  ,0=0.342;  kept  at  !7\=6600,  it  would  give  £=0.395. 

TO =440°,     7^  =  640°,    ,0=0.3628. 

A  tube  kept  at  the  temperature  !F0=4400  would  give  ,0=0.352; 
heated  wholly  at  the  temperature  7\  =  640°  it  would  give  ,0=0.401. 

7^=350°      ^  =  510°,    p= 0.245. 

A  tube  maintained  entirely  at  350°  would  have  given  p =0.234  ; 
at  510°  the  value  of  p  would  have  been  p= 0.398. 

In  all  these  experiments  the  selenium  had  been  placed,  at  the 
start,  in  the  extremity  of  the  tube  which  was  to  be  brought  to  the 
higher  temperature;  at  the  end  of  the  experiment  it  was  found 
collected  at  the  colder  end. 

In  spite  of  the  uncertainty  which  hangs  over  the  exact  values 
of  the  extreme  temperatures,  the  preceding  results  are  sufficiently 
conclusive. 


Decomposition 


Combination 


FIG.  131. 


r  o   TO  ~T 

FIG.  132. 


SECOND  CASE.  —  The  temperatures  of  the  various  parts  of  the  tube 
are  all  less  than  310°;  certain  of  these  parts  are  nevertheless  brought 
to  temperatures  higher  than  250°;  initially  the  system  does  not  con- 
tain hydroselenic  acid  ;  the  selenium  was  placed  in  the  part  of  the 
tube  which  is  to  be  heated  the  strongest. 

At  the  start,  the  representative  points  of  the  various  parts  of  the 
system  are  on  the  axis  OT  between  the  point  T0  and  7\  ;  those  points 
(Fig.  132)  between  C  and  T  are  in  the  region  of  combination;  among 
these  points  are  those  representing  the  parts  of  the  tube  where 


UNEQUALLY  HEATED  SPACES.  407 

selenium  is  collected  in  excess;  hydroselenic  acid  is  therefore 
formed,  x  increases  and  the  representative  straight  line  rises  until 
the  position  AB  is  reached,  at  the  moment  this  position  is  attained 
the  equilibrium  is  evidently  established  in  the  whole  system.  The 
composition  of  the  system  in  equilibrium  is  the  same  as  if  it  were  entirely 
brought  to  the  temperature  Tl  of  its  hottest  point ;  the  excess  of  selenium 
remains  at  the  hot  extremity  of  the  tube,  where  it  was  initially  put. 

One  may  say  that  in  the  case  at  hand  use  is  being  made  of 
Watt's  principle  reversed. 

Here  are  two  experiments  confirming  this  law: 

T0  =  laboratory  temperature,     7\  =  260°,    p  =  0.0312. 

A  tube  kept  during  the  same  time  wholly  at  the  temperature  7\= 
260°  gave  also  p= 0.0312. 

T0 = laboratory  temperature,     Tl  =  285°,    p = 0.084. 

A  tube  heated  throughout  at  the  temperature  7T1=285°  gave  ,0= 
0.085. 

THIRD  CASE. — The  cold  extremity  of  the  tube  is  at  a  temperature  less 
than  270°;  the  hot  end  is  at  a  temperature  higher  than  310°.  Initially 
the  tube  does  not  contain  hydro- 
selenic acid;  the  selenium  is  col- 
lected at  the  hot  end  of  the  tube. 
At  the  beginning  of  the  experiment  x' 
the  state  of  the  system  is  repre- 
sented by  the  straight  line  TQTl 
(Fig.  133),  of  which  one  part  C7\  is 
in  the  region  of  one  combination; 
certain  points  of  this  portion  repre- 
sent precisely  the  parts  of  the  tube 
where  the  selenium  is ;  in  these  parts 


O    To        C  Ta  T 

FIG.  133. 


selenhydric  acid  is  produced  and  diffuses  into  the  part  of  the  tube 
where  it  may  be  destroyed,  no  points  of  the  representative  line  being 
in  the  region  of  decomposition ;  x  increases,  therefore,  and  the  rep- 
resentative line  rises  to  the  position  AB,  where  it  touches  the  line 
DP  at  m. 

Will  it  stop  at  this  position? 


408  THERMODYNAMICS  AND  CHEMISTRY. 

The  warm  portions  of  the  tube  are  in  the  region  of  combination 
and  they  still  contain  selenium;  there  will  be  formed,  therefore,  in 
these  portions  selenhydric  acid,  and  x  will  increase,  which  will  bring 
the  representative  straight  line  to  A'B' ';  but  a  small  part  of  this 
line  np,  composed  of  points  whose  temperature  is  about  270°,  will 
be  in  the  region  of  decomposition;  in  the  corresponding  portions 
of  the  system  hydroselenic  acid  will  decompose;  selenium  will  be 
deposited,  having  been  transported  by  apparent  volatilization  from 
the  warm  regions  of  the  tube.  During  this  time  selenhydric  acid 
will  continue  to  be  formed  at  the  expense  of  the  selenium  contained 
in  the  warm  parts  of  the  tube.  It  is  clear  that,  in  order  for  equilib- 
rium to  be  established,  it  will  be  necessary  and  sufficient — 

1°.  That  all  the  selenium  be  transported  by  apparent  volatilization 
from  the  warm  portion  of  the  tube  whose  temperature  is  about  270°. 

2°.  That  the  repres  ntative  line  occupy  the  position  AB,  that  is 
to  say,  that  the  omposition  of  the  gaseous  mixture,  independent  of 
the  extreme  temperatures  TQ  and  T1}  is  the  same  as  in  a  system  kept 
wholly  at  the  temperature  270°. 

The  carrying  of  selenium  by  apparent  volatilization  into  the 
part  of  the  tube  whose  temperature  is  about  270°  may  be  masked 
by  the  real  volatalization  which  carries  selenium  from  the  warm 
parts  to  the  cold.  Ditte  and  Pelabon  devised  ingenious  experi- 
ments, but  too  long  to  describe  here,  to  separate  these  two  phe- 
nomena and  to  establish  the  first  beyond  dispute. 

As  to  the  second  law,  it  is  confirmed  by  the  following  experi- 
ments, due  to  Pelabon.  The  temperature  TQ  was,  in  all  the  ob- 
servations, the  laboratory  temperature;  the  temperature  Tt  varied 
greatly  without  p  changing  sensibly: 

7\  =  592°,        ,0=0.1986  after  160  hours 
7\  =  680°,        ,0  =  0.2002     "     162      " 
7\  =  700°,        ,0=0.1977    "       69      " 

Furthermore,  in  tubes  kept  entirely  at  270°,  after  heating 
periods  of 

192,.         288,  480,    and  490   hours, 

the  following  values  were  found  for  p: 

0.171        0.165        0.1605        0.163 


UNEQUALLY  HEATED  SPACES.  409 

302.  Phenomena  of  apparent  volatilization. — The  formation, 
at  a  high  temperature,  of  selenhydric  acid  from  selenium  and 
hydrogen  and  its  destruction  at  a  lower  temperature  produce  a 
transportation  of  selenium  by  apparent  volatilization. 

This  phenomenon  is  not  isolated;  in  conditions  similar  in  all 
points  to  the  preceding,  save  the  absolute  values  of  the  tempera- 
tures, which  are  here  higher,  Ditte  has  obtained  a  transportation 
of  tellurium  by  apparent  volatilization. 

Troost  and  Hautefeuille  *  likewise  discovered  certain  remark- 
able facts  concerning  apparent  volatilization. 

If  a  very  slow  current  of  silicon  tetrachloride,  SiCl4,  is  passed 
over  silicon  at  1200°  in  a  reverberating  furnace,  kept  quite  fixed 
at  this  temperature,  it  is  noticed  that  after  a  certain  time  silicon 
has  been  carried  by  apparent  volatilization  into  the  moderately 
warm  region  of  the  tube,  whose  temperature  is  comprised  between 
500°  and  800°.  In  reality,  by  the  action  of  the  silicon  tetrachloride 
on  silicon,  there  is  formed  silicon  trichloride,  Si2Cl<5,  volatile  at  these 
temperatures,  which  a  sudden  cooling  would  allow  to  be  collected, 
but  which,  as  we  have  seen,  completely  decomposes  at  tempera- 
tures included  between  700°  and  800°. 

Silicon  fluoride,  SiFl4,  passing  over  silicon  at  white  heat,  trans- 
ports the  latter  likewise,  by  apparent  volatilization,  into  the  parts 
of  the  tube  at  red  heat;  a  sudden  cooling  allows  collecting  the 
subfluoride  to  which  this  carrying  is  due. 

Chlorine,  passing  over  platinum  at  1400°,  carries  it,  by  apparent 
volatilization,  into  a  cooler  region  of  the  tube;  there  is  formed  a 
protochloride  of  platinum  which  a  sudden  cooling  allows  to  collect. 

Hydrogen,2  passing  at  red  heat  over  zinc  oxide,  kept  at  this 
temperature,  may  transform  it,  with  absorption  'of  heat,  into 
water  vapor  and  zinc  vapor;  inversely,  this  last  mixture  passing 
into  the  cooler  regions  of  the  tube,  is  transformed  anew,  with  libera- 
tion of  heat,  into  hydrogen  and  zinc  oxide  which  is  deposited  in 
the  crvstallized  state. 


1  TROOST  and  HAUTEFEUILLE,  Comptes  Rendus,  v.  73,  pp.  443  and  563, 
1871. 

8  H.  SAINTE-CLAIRE  DEVILLE,  Annales  de  Chimie  et  de  Physique,  3d  S,, 
v.  43,  p.  477,  1855. 


410  THERMODYNAMICS  AND  CHEMISTRY. 

By  an  analogous  phenomenon  of  mineralization*  hydrogen 
passing  over  amorphous  zinc  sulphide,  displaces  it  and  trans- 
forms it  into  hexagonal  crystals  of  wurtzite  (hexagonal  blende). 

303.  Vaporization  presents  phenomena  analogous  to  those 
just  studied. — The  study  of  simple  phenomena  of  vaporization 
furnishes  examples  of  each  of  the  three  cases  realized  by  Pelabon 
by  means  of  selenhydric  acid. 

At  the  temperatures  where  ordinarily  the  vaporization  of 
liquids,  such  as  water  and  alcohol,  is  observed  we  may  admit  that 
the  states  of  false  equilibria  are  so  close  to  the  states  of  veritable 
equilibria  that  they  are  practically  indistinguishable;  whence, 
whatever  are  the  temperatures  of  an  enclosure  containing  a  liquid 
and  its  vapor,  the  conditions  in  which  the  enclosure  exists  are  those 
of  the  first  case;  the  final  tension  of  the  vapor  in  the  enclosure  will 
be  the  same  as  if  the  enclosure  were  wholly  brought  to  the  tempera- 
ature  of  its  coldest  point;  it  is  in  the  cold  region  that  the  liquid 
will  be  found  wholly  condensed;  this  proposition  constitutes  one 
of  the  forms  of  Watt's  principle. 

The  condensation  of  phosphorus  vapor  to  the  state  of  white 
phosphorus  conforms  to  this  law;  it  is  not  the  same  with  the  con- 
densation of  phosphorus  vapor  to  the  state  of  red  phosphorus; 
at  the  ordinary  temperature  saturated  white  phosphorus  vapor 
has  a  tension  greatly  exceeding  the  tension  of  saturated  red  phos- 
phorus vapor,  the  latter  being  practically  zero;  nevertheless  the 
saturated  vapor  of  white  phosphorus  remains  indefinitely,  at  least 
in  the  dark  without  changing  into  red  phosphorus.  This  phe- 
nomenon of  false  equilibrium  makes  it  possible  to  predict  the  reali- 
zation of  an  enclosure  unequally  heated,  filled  with  phosphorus 
vapor  at  a  high  pressure,  and  in  which  phosphorus  would  condense, 
in  the  state  of  red  phosphorus,  otherwhere  than  on  the  coldest 
walls  of  the  enclosure ;  this  experiment  was  realized  by  Troost  and 
Hautefeuille; 2  it  is  the  more  convincing  as  the  condensation  of 
vapors  to  the  state  of  white  phosphorus,  modification  free  from 


1  H.  SAINTE-CLAIRE  DEVILLE  and  TROOST,  Annales  de  Chimie  et  de  Phy- 
sique, 4th  S.,  v.  5,  p.  118,  1865. 

2  TROOST  and   HAUTEFEUILLE,    Annales   de   I'Ecole  normale   superieure, 
2d  S.,  v.  2,  p.  253,  1873. 


UNEQUALLY  HEATED  SPACES.  411 

false  equilibrium,  takes  place,  at  the  same  time,  on  the  cold  walls 
of  the  tube. 

This  experiment  is  as  follows: 

White  phosphorus  is  heated  to  about  500°  in  the  middle  portion 
of  a  glass  tube  whose  two  ends  are  kept  the  one  at  350°  (vapor  of 
boiling  mercury)  and  the  other  at  324°  (vapor  of  mercury  bromide) . 
After  an  hour  and  a  half,  the  part  of  the  tube  at  350°  showed  an 
orange-red  coating,  uniform  and  translucent,  while  the  other  end, 
at  324°,  showed  not  the  least  trace  of  this,  but  instead  a  few  drops 
of  liquid  white  phosphorus. 

In  another  series  of  observations  Troost  and  Hautefeuille 
carried  one  of  the  ends  to  445°  (vapor  of  sulphur,  boiling  at  atmos- 
pheric pressure)  and  the  other  end  to  425°  (vapor  of  sulphur  boiling 
under  0.470  m.)  of  mercury;  after  fifteen  or  twenty  minutes  a 
good  red  coating  was  seen  in  the  end  at  445°,  and  at  the  most  a 
yellow  layer  extremely  thin  in  the  end  at  425°. 

The  transformation  of  the  vapors  of  cyanic  acid  into  solid 
cyamelide  gives  rise  to  analogous  observations,  likewise  due  to 
Troost  and  Hautefeuille. 

While  cyanic  acid  vapor  is  transformed  into  cyamelide  after 
several  hours  at  250°,  and  after  a  few  minutes  at  350°,  it  resists 
for  several  days  at  the  ordinary  temperature. 

If  cyanic  acid  vapor  be  introduced  into  an  enclosure  a  portion 
of  which  is  at  350°,  while  the  rest  is  maintained  at  100°,  this 
vapor  is  condensed  into  the  state  of  cyamelide  on  the  walls 
heated  to  350°,  and  the  tension  of  the  vapor  of  cyanic  acid  has 
the  value  of  1200  mm.,  which  is  that  of  saturated  vapor  of 
cyamelide  at  350°;  the  equilibrium  which  is  established  is  the 
same  as  if  the  enclosure  were  wholly  brought  to  the  temperature 
of  its  hottest  part. 


CHAPTER  XX. 
CHEMICAL  DYNAMICS  AND  EXPLOSIONS. 

304.  Chemical  dynamics.  —  Up  to  this  point  we  have  been  espe- 
cially occupied  with  the  conditions  in  which  a  chemical  system 
exists  in  the  state  of  equilibrium,  whether  it  be  a  question  of  verita- 
ble equilibrium  or  of  false  equilibrium  ;  we  have  treated  of  chemical 
statics. 

When  a  system  is  not  in  chemical  equilibrium,  it  is  trans- 
formed; its  true  condition  varies  from  one  instant  to  the  next; 
what  laws  govern  these  variations?  To  establish  these  laws  is  the 
object  of  chemical  dynamics,  the  part  of  chemical  mechanics  much 
less  advanced  than  statics. 

We  intend  to  indicate  here,  in  a  concise  manner,  some  of  the 
main  ideas  of  this  science;  in  order  not  to  enter  into  complications 
of  little  use,  we  shall  suppose  in  general,  and  except  as  noted  to 
the  contrary,  that  we  are  dealing  with  a  homogeneous  system. 

305.  Velocity  of  a  reaction.  —  To  be  definite,  let  us  suppose  that 
the  reaction  produced  in  the  system  studied  is  a  combination;    at 
a  certain  instant  t  the  system  contains  a  mass  m  of  the  compound 
formed  by  this  combination  ;   this  mass  increases  with  the  time,  so 
that  at  an  instant  t'  ,  later  than  t,  this  mass  has  a  value  m',  greater 
than  m. 

The  ratio  —-,  —  -  is  what  is  called  the  mean  velocity  of  the  com- 

L         6 

bination  between  the  instants  t  and  tf.     If  we  suppose  that  a  time 
if  nearer  and  nearer  to  t  is  taken,  (t'  —  t)  approaches  zero  and  simi- 

fyif  _  yft 

larly  for  (m'—m}',  but  the  ratio  —p  —  7-  approaches  a  limit  which 

t  —  t 

we  shall  denote  by  v: 


which  we  shall  call  the  velocity  of  combination  at  the  instant  t. 

412 


CHEMICAL  DYNAMICS  AND  EXPLOSIONS.  413 

306.  Fundamental    principle    of    chemical    dynamics.  —  The 

fundamental  principle  of  chemical  dynamics  is  the  following: 

The  velocity  of  the  combination  which  is  produced  at  a  given 
instant  within  a  homogeneous  system  is  determined  when  for  this 
instant  tl\e  nature  and  state  of  the  substances  forming  the  system  con- 
sidered, the  temperature  to  which  the  system  is  brought,  and  the  pres- 
sure it  supports  are  known. 

307.  Acceleration    of    a   reaction.  —  The    above   principle    is 
widely  different  from  the  one  which   underlies  the  dynamics  of 
local  motions  or  dynamics  properly  so  called;  in  order  to  indicate 
clearly  this  difference,  we  shall  introduce  a  notion  which  will  also 
be  useful  in  what  follows  ;  it  is  the  idea  of  acceleration  of  a  reaction. 

Let  v  be  the  velocity  of  reaction  at  the  instant  t,  and  vf  the 

•y'  _  -y 

velocity  at  the  instant  t',  later  than  t;  the  ratio  —f  —  -  is  the  mean 

t  —  t 

acceleration  of  the  combination  between  the  instants  t  and  t'  ;  if  we 
take   as    the   instant   t'    a  time  nearer  and  nearer  to  t,   (t'—t) 

approaches  0;  it  is  the  same  with  (v'—v);  but  the  ratio  -,  —  - 

t  —  t 

approaches  a  positive  or  negative  limit,  which  we  shall  denote  by  f, 

v'-v 


and  which  we  shall  call  the  acceleration  of  combination  at  the  instant  t. 
308.  Comparison  of  the  fundamental  principle  of  chemical 
dynamics  and  the  fundamental  principle  of  dynamics  properly  so 
called.  —  Let  us  take  now  a  material  point  M  which  moves  along  a 
straight  line;  let  I  be  the  distance  which,  at  the  instant  t,  this  point 
has  moved  from  a  certain  origin  0  ;  lf  be  the  distance  at  the  time  t'r 

later  than  t;  we  have  that  the  velocity  of  this  point  at  the  instant  t  is 

£/  _  j 
the  limit  approached  by  the  ratio  -7—^  as  t'  becomes  nearer  and 

nearer  to  t;  once  the  velocity  of  this  moving  point  is  denned 
at  the  instant  t,  the  acceleration  of  the  point  at  the  instant  t  is  de- 
nned in  exactly  the  same  way  as  we  have  denned  the  acceleration 
of  combination. 

For  the  particular  case  of  rectilinear  motion  to  which  we  shall 


414  THERMODYNAMICS  AND  CHEMISTRY. 

limit  ourselves  for  greater  simplicity,  the  fundamental  principle  of 
dynamics  is  the  following: 

//  m  is  the  mass  of  the  moving  point,  and  F  is  the  component,  taken 
along  a  straight  line,  of  the  force  acting  on  this  point,  the  acceleration 

F 

is  at  every  instant  equal  to  the  quotient  — : 


Now,  in  general,  the  force  F  is  known  when  the  nature  of  the  mass 
m,  its  position,  the  nature  and  position  of  the  substances  which  act 
on  this  mass,  are  known;  we  may  therefore  say  the  acceleration  is 
determined  when  the  state  and  circumstances  of  the  moving  body  are 
known. 

But  if  the  acceleration  is  determined  by  the  conditions  which 
determine  the  force,  it  does  not  follow  that  the  velocity  is  so  deter- 
mined; the  same  point,  movable  over  the  same  straight  line  and 
undergoing  the  influence  of  the  same  bodies,  may  pass  through  the 
same  position,  in  different  circumstances,  with  different  velocities. 

Suppose,  for  example,  that  there  is,  on  the  straight  line  con- 
sidered, a  position  of  equilibrium  for  the  point  undergoing  the 
action  of  the  forces  studied,  that  is  to  say,  a  position  where  the  point, 
acted  upon  by  these  forces,  would  remain  indefinitely  in  equilibrium 
if  it  were  placed  there  with  zero  velocity. 

It  may  very  well  happen  that  the  movable  point  arrive  in  this 
position  with  a  velocity  different  from  zero ;  then  it  will  not  remain 
there ;  it  will  go  beyond  by  virtue  of  the  velocity  acquired. 

Nothing  similar  to  this  can  occur  in  chemical  dynamics;  the 
system  being  placed  in  a  given  state  and  undergoing  definite  actions, 
the  reaction  of  which  it  is  the  seat  has  a  definite  velocity ;  in  particu- 
lar, if  the  system  is  placed  in  a  state  which  fulfils  the  equilibrium 
conditions,  the  velocity  of  reaction  is  necessarily  equal  to  zero;  if 
the  system  is  brought  in  any  way  whatever  into  this  state,  it 
remains  there  in  equilibrium ;  here  nothing  can  be  observed  analo- 
gous to  an  acquired  velocity. 

We  have  supposed  that  we  are  concerned  with  a  combination; 
but  all  we  have  just  said  may  be  applied  to  a  decomposition,  on  the 


CHEMICAL  DYNAMICS  AND  EXPLOSIONS.  415 

condition  of  denoting  by  m  the  mass  of  the  compound  which  has 
been  destroyed  or,  what  amounts  to  the  same  thing,  the  mass  of  the 
elements  this  destruction  has  set  free;  all  that  has  been  said  may 
likewise  be  applied  to  a  double  decomposition  or  to  any  other  com- 
plicated chemical  reaction,  on  the  condition  of  denoting  by  m  the 
mass  of  the  substance  or  substances  generated  by  this  reaction. 

309.  Influence  of  the  composition  of  the  system  on  the  velocity 
of  reaction. — Is  it  possible  to  further  specify  the  fundamental  prin- 
ciple of  chemical  dynamics  and  formulate  the  law  which,  for  a 
system,  joins  the  velocity  of  combination  to  the  conditions  in  which 
this  system  is  placed?    There  may  be  stated,  in  a  general  and 
certain  manner  only,  some  very  simple  propositions. 

Imagine,  in  the  first  place,  the  temperature  T  of  the  system  to 
be  kept  constant;  further,  that  either  the  pressure  it  supports  or 
the  volume  in  which  it  is  contained  is  kept  constant.  In  these  con- 
ditions we  may  state  the  following  proposition: 

The  velocity  of  reaction  diminishes  as  the  mass  m  of  the  substance 
or  substances  generated  by  this  reaction  increases. 

Another  proposition  may  be  joined  to  this  one: 

//  the  value  //  of  the  mass  m  corresponds  to  a  state. of  equilibrium 
of  the  system,  the  velocity  of  combination  approaches  zero  when  the 
mass  m  approaches  JJL. 

310.  Every  isothermal  reaction  is  a  moderated  reaction. — 
This  first  law  has  a  consequence  which  it  is  important  to  bring  out. 

Consider  a  reaction  which  is  produced  in  the  conditions  supposed : 
on  the  one  hand,  the  temperature  is  kept  constant;  on  the  other 
hand,  either  the  volume  or  the  pressure  is  kept  constant.  At  the 
instant  t,  m  is  the  mass  of  the  compound  formed,  and  v  the  velocity 
of  reaction;  at  the  instant  t',  later  than  t,  these  same  quantities 
have  the  respective  values  m'  and  vr.  The  system  having  been  the 
seat  of  a  reaction  in  the  direction  considered  between  the  instants 
t  and  t'j  the  mass  m'  is  necessarily  greater  than  m,  so  that  the 
velocity  t/  is  less  than  v. 

When  the  temperature  T  of  the  system  is  kept  constant  (isothermal 
reaction),  and  moreover  if  the  volume  it  occupies  or  the  pressure  it  sup- 
ports is  constant,  the  velocity  of  the  reaction  of  which  it  is  the  seat 
decreases  from  one  instant  to  the  next. 


416  THERMODYNAMICS  AND  CHEMISTRY. 

We  shall  call  moderated  reaction  one  whose  velocity  diminishes 
from  one  instant  to  the  next ;  we  may  then  say  that  every  isothermal 
reaction  accomplished  either  at  constant  pressure  or  at  constant  volume 
is  a  moderated  reaction. 

311.  The  acceleration  of  a  moderated  reaction  is  negative. — 
Let  v  be  the  velocity  of  a  reaction  at  the  instant  t;  let  v'  be  the 
velocity  of  the  same  reaction  at  an  instant  t'}  later  than  t'}  if  the 


reaction  is  moderated,  v'  is  less  than  v  and  the  ratio  —r — -,  mean 

t  —  * 

acceleration  of  the  reaction  between  the  instants  t  and  t',  is  negative ; 
this  is  true  however  near  tr  is  to  t,  which  fact  allows  to  say: 

A  reaction  which  is  moderated  at  the  instant  t  is  a  reaction  whose 
acceleration  is  negative  at  this  instant. 

312.  Influence  of  temperature  on  the  velocity  of  reaction. — 
Let  us  consider  a  combination,  which  at  the  temperature  of  the 
experiment  is  unlimited  and  does  not  cease  until  the  elements  con- 
tained in  the  system  are  entirely  combined ;  or  else  a  limited  com- 
bination, but  whose  limit,  in  the  conditions  of  operating  (either  at 
constant  volume  or  at  constant  pressure) ,  corresponds  to  a  value  fjL 
of  m}  value  independent  of  the  temperature. 

In  these  conditions  the  following  law  may  be  stated : 

Other  things  being  equal,  and  in  particular  the  composition  of  the 
system  corresponding  to  the  same  value  of  m,  the  velocity  v  of  combina- 
tion is  the  greater  as  the  temperature  is  higher. 

This  law  is  verified  a  fortiori  if  the  value  //  of  m  limiting  the 
combination  rises  with  the  temperature.  It  may,  on  the  contrary, 
become  inexact  if  the  value  of  fj.  decreases  while  the  temperature 
T  increases. 

Thus,  imagine  that  at  temperatures  T  and  T',  the  latter  being 
the  higher,  there  correspond  two  limiting  values  /*  and  //  of  m,  and 
that  p!  is  less  than  //.  At  the  temperature  T  a  system  for  which 
m  equals  //  is  the  seat  of  a  certain  reaction  whose  velocity  is  not 
zero;  on  the  contrary,  at  the  temperature  Tr  a  system  where  m  has 
the  value  //  is  in  equilibrium  and  the  reaction  there  has  zero 
velocity. 

313.  Example:  Phenomena  of  etherification. — A  great  number 
of  examples  illustrating  the  above  law  may  be  cited;   the  numer- 


CHEMICAL  DYNAMICS  AND  EXPLOSIONS.  417 

ous  investigations,  qualitative  and  quantitative,  which  have  been 
made  on  the  velocities  of  reaction  all  confirm  this  law;  we  shall 
mention  but  one  particularly  typical  example. 

The  etherification  of  an  alcohol  by  an  acid,  in  a  closed  vessel, 
hence  at  constant  volume,  reaches  a  limit  which  is  a  veritable 
equilibrium  state;  this  limit  corresponds  (Art.  179)  to  a  proportion 
of  ether  formed  which  is  independent  of  the  temperature,  so  that  the 
example  considered  is  within  the  conditions  for  which  our  law  was 
stated;  according  to  Berthelot1  the  velocity  of  this  reaction  is 
22,000  times  greater  at  +  200°  C.  than  in  the  neighborhood  of 
+  7°C. 

314.  Variation  of  the  velocity  due  to  a  small  change  of  com- 
position and  temperature. — Let  v  be  the  velocity  of  the  reaction 
considered,  when  the  mass  of  the  substance  or  of  the  collection 
of  substances  to  which  reaction  gives  rise  has  the  value  m  and  when 
the  temperature  has  the  value  T.  If  the  mass  m  underwent  alone 
a  small  variation  (m'  —  m),  the  velocity  would  undergo  an  increase 
A  (m'—m),  A  being  a  coefficient  whose  value  depends  on  the  mass 
m,  the  temperature  T,  and  the  other  conditions  in  which  the  system 
is;  an  increase  of  the  mass  m  always  produces,  other  things  being 
equal,  a  decrease  in  the  velocity  and  conversely;  it  follows  that  the 
coefficient  A  is  always  negative. 

If  the  temperature  T  underwent  alone  a  small  variation  (T'—T), 
the  velocity  would  undergo  an  increase  B  (T'—T),  where  B  is  a 
coefficient  depending  on  the  mass  m,  the  temperature  T,  and  the 
other  conditions  of  the  system;  we  know  that,  if  the  reaction  is 
unlimited,  or  if  the  value  /JL  of  m  which  limits  the  reaction  does  not 
decrease  as  the  temperature  increases,  a  rise  of  temperature  pro- 
duces, other  things  being  equal,  an  increase  in  the  velocity  of  re- 
action ;  in  these  conditions  the  coefficient  B  is  positive. 

Suppose  in  particular  that  the  reaction  studied  obeys,  as  it  so 
often  happens,  those  laws  (Arts.  287  to  292)  whose  existence  was 
illustrated  by  the  reaction 


1  BERTHELOT,  Essai  de  Mecanique  chimique  fondee  sur  la  Thermochimie, 
v.  2,  p.  93. 


418  THERMODYNAMICS  AND  CHEMISTRY. 

or  by  the  reaction 

3SiCl4  +  Si  =  2Si2Cl6, 

as  was  shown  in  the  preceding  chapter;  as  long  as  the  tempera- 
ture is  less  than  that  we  have  denoted  by  6,  and  starting  from 
which  begins  the  region  of  veritable  equilibria,  the  reaction  which 
may  be  produced  in  the  system  is  either  limited  by  a  value  of 
fi  which  increases  with  the  temperature  (temperatures  less  than  r) 
or  else  practically  unlimited  (temperatures  included  between  r  and 
6)  .  Therefore,  for  reactions  which  follow  the  laws  of  which  the  forma- 
tion of  hydrogen  sulphide  is  the  type,  the  coefficient  B  is  surely  positive 
at  temperatures  below  the  point  6. 

If  simultaneously  an  increase  of  (m'—m)  is  imposed  upon  the 
mass  m  and  (T'—T)  upon  the  temperature  T,  the  velocity  v  under- 
goes the  increase 

(1)  v'-v  =  A(m'-m)+B(T'-T), 

the  sum  of  the  two  partial  increments  of  which  we  have  just  spoken. 
315.  Return  to  isothermal  reactions.  —  Suppose  first  the  tem- 
perature kept  constant  or  the  reaction  isothermal;  suppose  that  T, 
m,  v  refer  to  the  instant  t,  and  T",  m',  v'  to  the  temperature  t', 
later  than  t  and  very  close  to  t;  we  shall  have 


Then  by  definition 

m'-m=v(t'-t) 
and 

v'-v=f(t'-t). 

Equation  (1)  then  becomes 

(2)  T=Av. 

We  have  seen  that  A  is  always  negative;  it  is  the  same  with  7% 
as  we  have  already  found. 

316.  Adiabatic  reactions.  —  Suppose  now  that  while  the  reaction 
lasts  there  is  no  interchange  of  heat  between  the  system  and  sur- 


CHEMICAL  DYNAMICS  AND  EXPLOSIONS.  419 

rounding  bodies;  the  reaction  is  then  adiabatic;  this  supposition  is 
approximately  verified  for  very  rapid  reactions;  the  temperature 
T  varies  from  one  instant  to  another  in  the  same  time  as  the  mass 
m;  it  is  easy  to  determine  the  law  of  this  variation. 

Between  neighboring  instants  t  and  tf  there  is  formed  in  the 
system  a  mass  (mf  —  m)=v(t'  —  t)  of  the  substance  or  collection  of 
substances  produced  by  the  reaction ;  if  this  formation  takes  place 
at  constant  temperature,  it  will  be  accompanied  by  a  liberation  of 
heat. 

L(m'-m)=Lv(t'-t), 

L  being  the  heat  called  out  by  the  reaction  which,  for  the  condition  in 
which  the  system  is  placed,  would  trans,  orm  one  gramme  of  matter. 

Furthermore,  the  temperature  of  the  system  changes  from  T  to 
T' ;  if  this  modification  were  produced  alone  it  would  absorb  a 
quantity  of  heat  C(T'  —  T},  where  C  is  the  total  heat  capacity  of  the 
system  for  the  conditions  in  which  it  is  placed. 

The  total  heat  set  free  by  the  system  has  the  value 

Lv(t'-t)-C(T'-T). 

This  liberation  of  heat  being  zero  for  an  adiabatic  transforma- 
tion, we  have 

(3)  T'-T=^v(t'-t). 

Substituting  this  value  of  T' -T  in  equation  (1),  and  noting  also 
that  by  definition 


m'-m=v(t'-t),        i/-v=r(t'-t), 


we  find 
(4) 


The  ratio  of  the  acceleration  to  the  velocity  has  not  the  same 
value  in  an  adiabatic  reaction  as  in  an  isothermal  reaction. 


420  THERMODYNAMICS  AND  CHEMISTRY. 

317.  An  adiabatic  reaction  may  have  a  positive  acceleration. 

— It  may  even  happen  that  the  acceleration  of  the  reaction,  neces- 
sarily negative  for  an  isothermal  reaction,  is  positive  for  an  adia- 
batic reaction. 

Let  us  consider  in  particular  a  reaction  following  the  law  of 
which  the  formation  of  hydrogen  sulphide,  studied  in  the  last 
chapter,  is  the  type,  and  let  us  suppose  the  temperature  is  less  than 
the  temperature  0,  above  which  veritable  equilibrium  states  may 
be  observed. 

From  what  has  been  said  previously,  B  is  then  positive ;  as  we 
have  already  indicated  in  the  preceding  chapter  (Art.  293),  the  only 
reaction  which  can  be  observed  is  exothermic,  so  that  L  is  positive ; 

DT 

C  is  also  positive,  hence  the  ratio  — ^-  is  assuredly  positive.     The 

0 

ratio  —  has,  for  an  adiabatic  modification,  a  value  (  A  -f  — — ) ,  which 
v  \         C  / 

is  surely  greater  than  the  value  A  of  the  same  ratio  for  an  isothermal 
transformation;  this  latter  is  necessarily  negative,  but  it  may 
happen  that  the  first  is  positive. 

318.  Reactions  with  positive  acceleration  and  explosive  reac- 

( 

positive? 

The  acceleration  being  positive,  the  velocity  of  the  reaction  will 
increase  with  the  time;  at  the  same  time,  as  shown  in  equation  (3), 
where  L  and  C  are  positive,  the  temperature  will  rise  from  one 
instant  to  the  next. 

It  may  be  said  that  these  are  the  two  characteristics  by  which 
explosive  reactions  are  recognized ;  or  rather,  chemists  generally  con- 
sider explosive  a  reaction  which  has  these  two  characteristics  in  a 
very  high  degree;  but  to  give  the  expression  explosive  reaction  a 
clear  definition,  we  shall  henceforth  agree  that  it  denotes  a  reaction 
whose  acceleration  is  positive  and  which  is  accompanied  by  an  eleva- 
tion of  temperature  in  the  system. 

We  may  then  state  the  following  proposition: 

While  an  isothermal  reaction  is  forcibly  a  moderated  reaction,  an 
adiabatic  reaction  may  be  explosive. 


IS 


CHEMICAL  DYNAMICS  AND  EXPLOSIONS.  421 

319.  Condition  in  order  that  an  adiabatic  reaction  be  explo- 
sive. —  In  particular,  a  reaction  of  the  type 


adiabatic  and  produced  at  a  temperature  lower  than  the  point  6,  where 
states  of  veritable  equilibria  begin  to  manifest  themselves,  is  explosive 

r>T 

if  the  essentially  positive  ratio  -  -  is  greater  than  the  absolute  value 

o 

of  the  negative  coefficient  A. 

320.  Indetermination  of  the  temperature  which  renders  a 
reaction  explosive.  —  It  is  evident  from  the  above  that,  in  order  to 
decide  if  a  reaction  is  or  is  not  explosive,  it  does  not  suffice  to  indi- 
cate the  composition  of  the  system,  the  temperature  and  the 
pressure  (or  volume)  ;  all  these  conditions  remaining  the  same,  it 
may  happen  that  the  same  reaction  is  or  is  not  explosive  according 
to  the  law  which  rules  the  variations  of  temperature  ;   a  reaction, 
explosive  if  the  system  is  enclosed  in  a  vessel  impermeable  to 
heat,  becomes  moderated  if  it  is  rendered  isothermal. 

This  explains  why  various  observers,  operating  by  different 
methods,  may  give  very  different  indications  concerning  the  con- 
ditions in  which  a  reaction  becomes  explosive;  Mitscherlich  found 
that  the  combination  of  oxygen  and  hydrogen  became  explosive 
at  674°;  Mallard  and  Le  Chatelier  have  indicated  for  this  phe- 
nomenon the  temperature  of  about  550°;  A.  GautierandH.  Helier, 
by  heating  the  mixture  of  oxygen  and  hydrogen  in  a  porcelain 
vessel  filled  with  porcelain  chips  which  increased  the  heating  surface 
and  rendered  the  reaction  almost  isothermal,  were  able  to  retard 
until  845°  the  temperature  at  which  the  formation  of  water  becomes 
explosive. 

In  the  future,  when  we  speak  of  the  conditions  in  which  a 
reaction  becomes  explosive,  we  shall  always  suppose  the  system 
placed  in  an  enclosure  impermeable  to  heat,  so  that,  the  reaction  is 
adiabatic. 

321.  Stability  and  instability  of  limiting  false  equilibria.  — 
Related  to  the  questions  we  have  just  examined  is  the  study  of  the 
stability  of  limiting  false  equilibria,  as  we  shall  see.    This  question 
may  be  treated  in  a  very  general  manner;  but  in  view  of  the 


422 


THERMODYNAMICS  AND   CHEMISTRY. 


applications  we  wish  to  make  of  it,  it  will  be  sufficient  to  discuss  a 
reaction  of  the  type 


To  be  definite,  we  shall  suppose  there  is  question  of  a  combination. 
The  system  will  be,  for  example,  treated  at  constant  pressure. 

Take  the  temperature  T  as  abscissae  (Fig.  134),  and  for  ordinates 
the  ratio  x  between  the  mass  m  of  the  compound  contained  by  the 

system  and  the  mass  m  of  the  same 
compound  which  it  would  contain  if 
the  compound  were  pushed  as  far  as 
possible.  From  this  definition 

m  =  Mx. 

The  region  of  false  equilibria  is 
separated  from  that  of  combination 
by  a  line  FF'  ,  which  rises  from  left 
to  right;  the  various  points  of  this 
line  represent  the  limiting  states  of 
false  equilibrium  of  the  system. 
Let  /  and  /'  be  two  points  close  together  taken  on  the  line  FF'  ; 
T  and  x  are  the  coordinates  of  the  first,  T'  and  x'  of  the  second. 
The  velocity  v  of  the  combination  approaches  zero  while  the  system 
approaches  the  state  represented  by  the  point  /;  it  also  approaches 
zero  when  the  system  approaches  /';  the  difference  (v'  —  v)  relative 
to  these  two  states  is  therefore  equal  to  zero,  which  may  be  written, 
in  accordance  with  equation  (1), 

A(m'-m)+B(T'-T)  =  0, 


or,  since  m  =  Mx,    mf  = 

AM(x'-x)+B(T'-T)=Q. 
Now  in  the  right  triangle  /</>/'  we  have 


consequently 


~. 


CHEMICAL  DYNAMICS  AND  EXPLOSIONS.  423 

By  the  preceding  equation  this  becomes 

(5)  tan  /'/<£=  - 


But  the  point  f  being  very  near  to  /,  the  line  //'  becomes  identical 
with  the  straight  line  fd  which  touches  at  /  the  line  FF'  ',  and  the 
tangent  ff</)  is  what  we  have  called  (Art.  146)  ^trigonometrical 
tangent  at  f  of  the  line  FF'  ;  we  see  that  the  trigonometrical  tangent 

D 

has  the  value  —  -..    This  result  will  be  useful  to  us  shortly. 


322.  Every  state  of  false  equilibrium  which  is  not  limiting  is 
indifferent.  —  Let  us  take  a  case  of  false  equilibrium  and  ask  our- 
selves if  this  state  of  false  equilibrium  is  stable,  indifferent,   or 
unstable. 

Suppose  in  the  first  place  that  the  state  of  false  equilibrium  con- 
sidered is  not  a  case  of  limiting  false  equilibrium;  the  point  repre- 
senting it  is  in  the  interior  of  the  region  of  false  equilibria  and  not  on 
the  limiting  line;  give  the  system  a  slight  change,  corresponding  to 
small  variations  of  T  and  x;  we  may  always  take  these  variations 
small  enough  so  that  the  state  of  the  deranged  system  is  still  repre- 
sented by  a  point  in  the  region  of  false  equilibria,  c*se  in  which  the 
deranged  system  will  be  still  in  equilibrium;  we  may  therefore  state 
the  proposition: 

Every  state  of  false  equilibrium  which  is  not  a  state  of  limiting 
false  equilibrium  is  a  state  of  indifferent  equilibrium. 

323.  If  the  temperature  is  constant,  every  false  equilibrium 
is  stable.  —  Take  now  a  limiting  state  of  false  equlibrium,  represented 
by  a  point  /  on  the  line  FF',  and  impose  a  slight  change  upon  the 
system;   it  may  happen  that  this  change  causes  the  point  repre- 
senting the  state  of  the  system  to  penetrate  into  the  interior  of  the 
region    of   false   equilibria;  for  such  a  derangement  the  system 
is  surely  in  indifferent  equilibrium  and  we  have  no  need  to  concern 
ourselves  further  with  such  a  change;    it   may  happen,  on  the 
contrary,  that  this  change  brings  the  point  representing  the  system 
to  g  (Fig.  135)  in  the  region  of  combination,  and  this  is  the  case 
we  are  going  to  discuss. 

Suppose,  in  the  first  place,  that  after  having  slightly  deranged 


424 


THERMODYNAMICS  AND  CHEMISTRY. 


the  system,  it  is  placed  in  such  conditions  that  it  can  no  longer 
undergo  other  than  isothermal  changes;  the  point  g  representing 

the  state  of  the  system  being  in 
the  region  ois  combination,  x  will 
increase  without  T  varying;  the 
representative  point  g  will  rise 
along  a  parallel  to  Ox  through 
the  point  g,  and,  as  this  last  point 
is  assuredly  beneath  the  line  FF', 
the  point  representing  the  state 
of  the  system  will  approach  this 
line;  the  reaction  will  tend  to 
f  bring  back  the  system  to  the 
equilibrium  state.  A  state  of  limit- 
ing false  equilibrium  is  a  state  of  stable  equilibrium  for  a  system, 
which  once  deranged  can  no  longer  undergo  other  than  isothermal 
transformations. 

324.  If  the  reactions  are  all  adiabatic,  the  limiting  false  equi- 
libria may  be  stable  or  unstable. — Suppose,  now,  that  the  system 
once  deranged  can  undergo  only  adiabatic  transformations.  The 
representative  point  is  at  g  at  the  instant  t;  let  g'  be  the  position 
it  occupies  at  the  instant  t' ',  near  to  t  but  later  than  t:  let  T  and  x 
be  the  coordinates  of  the  point  g,  and  T ',  x'  the  coordinates  of  the 
point  (f . 

We  shall  have  m/  —  m= M(x' — x) , 


T    T 
FIG.   135. 


and  as 


m'-m=v(t'-t), 


we  shall  have  x'—x= -^(f  —  t) . 

Further,  the  transformation  being  adiabatic,  (T'  —  T)  is  given 
by  equation  (3): 


(3) 


In  the  triangle  gfg'  we  have 
gr=T'-T, 


CHEMICAL  DYNAMICS  AND  EXPLOSIONS.  425 

consequently 


We  may  therefore  write  the  following  equality: 


Let  fd  be  the  line  tangent  at  /  to  the  curve  //';  we  have  seen  that 
(5)  tan(/0,OT)  =  --A. 

These  two  results  allow  us  to  discuss  the  stability  or  instability 
of  one  state  of  equilibrium. 

According  to  equation  (3),  where  L,  C,  v,  (t'-t)  are  positive 
quantities,  (T'  —  T]  is  positive;  the  point  tf  is  to  the  right  of  the 
point  g  on  the  line  whose  tangent  is  given  by  equation  (5)  ;  there 
are  then  two  principal  cases  to  distinguish: 

1°.  We  have  the  equation 


m  -.>.._. 

^     AM' 


The  line  go/  rises  from  left  to  right  more  rapidly  than  the  line 
fd  ;  the  representative  point  of  the  system  which,  on  account  of  the 
reaction,  moves  from  left  to  right  on  the  line  go/,  approaches  the 
line  fO,  tangent  to  the  line  FF'  ;  and  since  over  a  short  distance  a 
line  may  be  considered  identical  with  its  tangent,  the  point  repre- 
senting the  system  approaches  the  limiting  line  of  the  false  equilib- 
ria; the  system  tends  to  resume  a  state  of  equilibrium. 

A  system  compelled,  after  being  deranged,  to  undergo  only  adia- 
batic  modifications  is  iz>  stable  equilibrium  in  a  state  of  limiting  false 
equilibrium  if  condition  (7)  is  verified. 

2°.  We  have  the  equation 

C  B 

<  ~ 


LM        AM' 


426 


THERMODYNAMICS  AND  CHEMISTRY. 


By  similar  reasoning,  it  is  found  that  the  adiabatic  reaction  has 
the  same  effect  of  separating  farther  the  representative  point  from 
the  line  FFf '. 

A  system  compelled,  after  being  deranged,  to  undergo  only 
adiabatic  changes  is  in  unstable  equilibrium  in  a  state  of  limiting 
false  equilibrium  if  condition  (8)  is  verified. 

The  line  FF'  rises  from  left  to  right  (Fig.  136),  rapidly  at  first, 


OF  r          T 

FIG.  136. 

then  more  and  more  gradually;  at  the  temperature  which  in  the  pre- 
ceding chapter  (Art.  288)  we  have  denoted  by  r  it  becomes  sensibly 
tangent  at  P  to  the  straight  line  AA',  parallel  to  OT ',  and  whose 
constant  ordinate  is  equal  to  1.  At  this  moment  its  tangent  is 
very  nearly  0;  therefore  for  limiting  false  equilibria  relative  to 

D 

temperatures  close  to  r  but  less  than  r,  —  -J-T>  has  very  small  posi- 
tive values. 

On  the  other  hand,  there  is  no  reason  why  the  positive  ratio  j-^ 

JuivL 

should  assume,  in  the  neighborhood  of  the  states  of  false  equilibrium 
in  question,  a  very  small  value;  the  inequality  (7)  will  therefore  be 
verified  for  these  states  of  false  equilibrium;  the  states  of  limiting 
false  equilibrium  which  are  sufficiently  near  to  the  point  P  are  assuredly 
stable  even  for  a  system  enclosed  in  a  covering  impermeable  to  heat. 

Is  it  the  same  for  all  the  limiting  states  of  false  equilibrium, 
represented  by  the  various  points  of  the  line  FF't  It  may  be  that 
this  is  so;  it  may  be,  on  the  contrary,  that  there  exists  on  the  line 
FF'  a  point  e  where  we  would  have  the  equation 


(9) 


LM 


B 

AM 


CHEMICAL  DYNAMICS  AND  EXPLOSIONS. 


427 


In  this  case  the  limiting  false  equilibria  represented  by  the 
various  points  of  the  line  eP  would  be  stable  in  a  system  maintained 
in  an  enclosure  impermeable  to  heat,  while,  in  the  same  circumstances, 
the  limiting  false  equilibria  represented  by  the  various  points  of  the  line 
Fs  would  be  unstable. 

325.  Relation  between  the  limiting  false  equilibria  which  are 
unstable  and  the  explosive  reactions. — The  letter  A  which  figures 
in  equations  (7)  and  (8)  represents  a  negative  quantity;  B,  on  the 
contrary,  for  the  reactions  we  studied,  is  positive;  as  to  the  letters 
M,  C,  L,  they  represent  quantities  essentially  positive;  it  is  then 
evident  that  equation  (7)  may  be  written 


(7') 


while  equation  (8)  may  be  expressed  as 


(80 


We  are  therefore  led  to  the  following  conclusion: 

A  system  is  kept  in  an  enclosure  impermeable  to  heat;  every  point 
situated  in  the  region  of  combination  and  near  to  a  part  of  the  line 
FF'  which  represents  stable  limiting  states  -figures  a  state  where  the 
system  is  the  seat  of  a  moderated  combination;  every  point  situated  in 
the  region  of  combination  and  near  to  a  point  of  the  line  FF'  which 
represents  unstable  states  figures  a  state  where  the  system  is  the  seat  of 
an  explosive  combination. 


Moderated 

Combination^" 


FIG.  137. 


326.  Three  cases  to  distinguish. — Consider  in  the  first  place  the 
case  where,  on  the  line  FF'  (Fig.  137),  there  exists  a  point  e  sepa- 


428 


THERMODYNAMICS  AND  CHEMISTRY. 


rating  the  limiting  stable  states,  represented  by  the  various  points 
of  eP,  from  the  limiting  unstable  states  represented  by  the  various 
points  of  Fe.  In  this  case  there  exists  surely  a  line  ee',  starting 
from  the  point  e,  and  dividing  the  region  of  combination  into  two 
subregions.  Every  point  of  the  subregion  located  above  ee' 
represents  a  state  where  the  system,  enclosed  in  an  envelope 
impermeable  to  heat,  is  the  seat  of  a  moderated  combination; 
every  point  of  the  subregion  situated  below  ee'  represents  a  state 
where,  in  the  same  circumstances,  the  system  is  the  seat  of  an 
explosive  combination. 

If,  on  the*  contrary,  the  line  FF'  has  no  point  such  as  e,  all  the 
states  of  this  line  represent  states  of  limiting  equilibrium  which  are 
stable  if  the  system  is  in  an  enclosure  impermeable  to  heat ;  this 
line  is  confined,  therefore,  throughout  its  length  to  the  region  of 
moderated  combination. 

There  are  then  two  cases  to  distinguish : 

In  the  first  case  (Fig.  138)  there  exists  a  line  i}tf  which  divides 

x 
A 


Explosive 
Combinations 


FIG.  138. 

the  region  of  combination  into  two  subregions:  a  subregion  of 
moderated  combination  situated  between  T?T/  and  FP,  and  a  sub- 
region  of  explosive  combination  situated  below  T?)/. 

In  the  second  case  every  combination  produced  within  the  sys- 
tems enclosed  in  a  vessel  impermeable  to  heat  is  a  moderated 
combination. 

Return  to  the  case  represented  in  Fig.  137. 

Take  in  this  case  a  system  which  contains  the  elements  suitable 
to  form  a  compound,  but  which  contain  no  trace  of  the  compound ; 
x  is  equal  to  0,  and  the  representative  point  of  the  state  of  the  system 
lies  on  the  straight  line  OT. 


CHEMICAL  DYNAMICS  AND  EXPLOSIONS. 


429 


As  long  as  the  temperature  is  less  than  OF  the  system  is  in  the 
state  of  false  equilibrium;  at  the  moment  the  temperature  attains 
the  value  OF,  which  is  the  reaction-point  of  the  system,  a  combina- 
tion is  produced ;  if  the  system  is  in  an  enclosure  impermeable  to  heat, 
this  combination  is  explosive. 

327.  The  reaction-point  of  a  mixture  is  in  general  below  the 
explosion-point. — We  cannot  cite  with  certainty  any  homogeneous 
system  having  these  properties;  for  all.  the  temperature  of  the 
reaction-point  is  well  below  that  for  which  the  reaction  may  become 
explosive;  thus,  according  to  A.  Gautier  and  Helier,  the  reaction- 
point  of  a  mixture  of  oxygen  and  hydrogen  does  not  exceed  180°, 
while  the  explosive  formation  of  water  vapor  has  never  been  ob- 
served at  less  than  500°.  Further,  it  has  been  known  a  long  time  1 
that  the  mixtures  of  formene  and  oxygen,  carbon  bisulphide  and 
oxygen,  chlorine  and  hydrogen,  at  temperatures  included  between 
350°  and  500°,  unite  slowly  without  explosion;  they  are  therefore 
as  many  mixtures  for  which  the  point  e  does  not  exist. 

These  various   mixtures  therefore  form  systems  for  which  it 
is  proper  to  employ  Fig.  138  or, 
what  amounts  to  the  same  thing, 
Fig.  139. 

Given  the  composition  x  of 
such  a  system ;  draw  a  parallel  to 
OT  whose  constant  ordinate  is 
Ox=x;  this  straight  line  inter- 
sects the  line  FP  in  a  point  /, 
of  abscissa  t,  and  the  line  TJTJ' 
in  a  point  e  of  abscissa  T. 

At  temperatures  less  than  t  the  mixture  of  composition  x  is  in 
the  state  of  false  equilibrium. 

At  temperatures  above  t,  reaction-point  of  the  mixture,  a  com- 
bination is  produced.  If  the  temperature  is  comprised  between  t 
and  T,  the  combination  is  moderated,  even  if  the  system  is  enclosed 
in  a  covering  impermeable  to  heat.  At  temperatures  above  T  the 
combination  is  explosive,  provided  the  system  is  enclosed  in  a 


x 


P    A' 


t     -n     T 
FIG.  139. 


1  A.  GAUTIER,  Bull,  de  la  Soc.  chimique  de  Paris,  v.  13,  p.  1,  1869;  J.  H. 
VAN'T  HOFF  and  V.  MEYER  later  confirmed  these  observations. 


430  THERMODYNAMICS  AND  CHEMISTRY. 

non-conducting  envelope;  the  temperature  T  may  be  called  the 
explosive  temperature  of  the  mixture  which  contains  a  proportion 
x  of  the  compound  substance;  the  line  f]t\'  is  the  line  of  tempera- 
tures of  explosion  of  the  system  taken  in  the  given  conditions, 
either  volume  or  pressure  constant. 

328.  The  interval  between  these  two  points  and  safety  explo- 
sives. —  The  explosive  temperature   I7  of  a  mixture  exceeds  the 
reaction-point  t  of  the  same  mixture;   the  interval  between  these 
two  temperatures  may  be  very  great;   this  is  true,  according  to 
Mallard  and  Le  Chatelier,1  for  mixtures  of  oxygen  and  methane; 
also  it  is  possible  in  coal-mines  to  employ  explosives  which  produce 
a  temperature  higher,  it  is  true,  than  the  reaction-point  of  these 
mixtures,  but  below  their  explosive  temperature;  these  explosives 
cannot  cause  fire-damp  mixtures  to  detonate. 

329.  Mixtures  which  are  never  detonating.—  It  may  happen 
that  the  system  studied  does  not  admit  a  line  of  temperatures  of 
explosion,  and  that  the  combination  in  it  is  always  moderated,  even 
in  an  envelope  impermeable  to  heat  ;  this  is,  for  instance,  the  case 
for  the  combination  of  hydrogen  and  sulphur,  of  which  we  have 
spoken  at  length  in  the  preceding  chapter  (Art.  287). 

330.  Explosive  combinations.  —  All  we  have  just  said  on  the 
subject  of  a  system  in  which  an  exothermic  compound  is  formed, 
according  to  the  laws  of  which  the  reaction 


has  given  us  an  example,  may  be  repeated  mutatis  mutandis 
regarding  a  system  in  which  an  endothermic  system  is  destroyed, 
according  to  the  laws  of  which  the  reaction 

2Si2Cl6  =  3SiCl4  +  Si 

has  furnished  us  the  type  (Chap.  XVIII,  Art.  292). 

Here  again  all  the  cases  studied  experimentally  seem  to  range 
themselves  in  two  classes. 

In  certain  systems  the  destruction  of  the  compound  substance 
is  always  a  moderated  reaction,  even  when  this  reaction  is  adia- 
batic. 

1  Commission  des  Substances  explosives:  Sous-commission  speciale  (E. 
MALLARD,  rapporteur)  (Annales  des  Mines,  8th  S.,  v.  14,  p.  197,  1888). 


CHEMICAL  DYNAMICS  AND  EXPLOSIONS. 


431 


In  others,  on  the  contrary,  there  exists  a  line  Tft'  (Fig.  140)  of 
explosive  temperatures  which  divides  the  region  of  decomposition 
into  two  regions: 

1°.  A  region  of  moderated  decomposition  having  the  form  of  a 
band  more  or  less  wide  between  the  lines  T}-if  aad  FP-, 

2°.  A  region  of  explosive  decomposition,  situated  above  TJTJ'. 

That  such  a  disposition  is  indeed  that  proper  to  the  Beater 
number  of  explosive  substances  is  what  chemists  accustomed  to 
their  manipulation  have  found.  ''Below  the  temperature  at  which 
they  become  explosive,"  says  Berthelot,1  "and  during  an  interval 


FIG.  140. 

of  temperature  more  or  less  extended,  all  the  exothermic  decomposi- 
tions must  be  produced  in  a  progressive  manner."  "  Certain  explo- 
sive matters  are  sometimes  decomposed  very  slowly, 'starting  from 
room  temperature,  and  produce  detonations  only  if  the  temperature 
is  raised  intentionally  or  by  accident."  2 

331.  Influence  of  pressure  on  the  point  of  explosion. — All 
we  have  said,  all  the  figures  we  have  drawn,  suppose  the  system 
heated  either  at  constant  pressure  or  at  constant  volume;  to  fix 
our  attention,  suppose  it  is  question  of  a  system  heated  at  constant 
pressure;  the  form  and  position  of  the  lines  yr)',  FP  depend  on 
the  value  of  this  constant  pressure  and  change  with  this  value;  in 
particular  it  may  happen  that  the  line  TJTJ'  exists  when  the  value 
of  the  pressure  is  taken  within  a  certain  interval,  and  no  longer 
exists  when  it  is  taken  within  another  interval ;  under  all  the  pres- 

1  BERTHELOT,  Essai  de  mecanique  chimique,  fondee  sur  la  Thermody- 
namique,  v.  2,  p.  66. 

8  BERTHELOT,  Sur  la  force  des  matieres  explosives,  v.  2,  p.  71. 


432  THERMODYNAMICS  AND  CHEMISTRY. 

sures  of  the  first  interval,  the  system  may  be  the  seat  of  an  explosive 
reaction;  on  the  contrary,  underpressures  belonging  to  the  second 
interval  there  will  never  occur  other  than  a  moderated  reaction. 

Within  ozonized  oxygen,  under  ordinary  pressure,  the  ozone 
undergoes  moderated  decomposition  from  the  temperature  of  the 
laboratory;  but  at  no  temperature  does  this  decomposition  become 
explosive;  when,  on  the  contrary,  ozonized  oxygen  is  strongly 
compressed,  the  ozone  may  decompose  with  explosion.1 

Hydrogen  arsenide  is  an  endothermic  compound  which  is 
slowly  destroyed  at  room  temperature,  and  which  under  atmos- 
pheric pressure  does  not  become  detonating  at  any  temperature, 
not  even  that  developed  by  the  electric  spark;  if  one  detonates  in 
hydrogen  arsenide  a  fulminate-of-mercury  cap,  the  gas  undergoes 
not  only  a  great  rise  in  temperature,  but  also  an  energetic  com- 
pression, and  it  decomposes  with  explosion.2 

This  influence  of  pressure  on  the  possibility  of  explosion  has  been 
clearly  shown  by  Berthelot  and  Vielle  3  in  their  studies  on  acety- 
lene. 

Acetylene  is  an  endothermic  compound  which,  according  to  the 
principle  of  the  displacement  of  equilibrium  by  variation  of  tem- 
perature, is  formed  directly  at  the  very  high  temperature  of  the 
electric  arc  (Art.  177). 

Liquid  acetylene  is  an  explosive  compound  whose  effects  are 
of  the  order  of  those  produced  by  guncotton. 

The  same  is  not  true  of  acetylene  taken  in  the  gaseous  state 
under  atmospheric  pressure;  in  these  conditions  this  gas  does  not 
detonate  either  by  the  action  of  a  red-hot  platinum  wire  or  by  an 
electric  spark,  not  even  by  the  explosion  of  a  fulminating  mercury 
cartridge. 

Under  a  pressure  of  two  atmospheres,  on  the  contrary,  acetylene 
gas  behaves  like  an  explosive  compound;  it  decomposes  with 


1  HAUTEFEUILLE  and  J.  CHAPPUIS,  Comptes  Rendus,  v.  91,  p.  522,  1880. 

*  BERTHELOT,  Sur  la  force  des  matieres  explosives,  v.  i,  p.  114. 

1  BERTHELOT  and  VIELLE,  Annales  de  chimie  et  de  physique,  7th  S.,  v.  n, 
1897.  There  is  a  good  resum6  of  the  explosive  properties  of  acetylene  by 
L.  MARCHIS,  Lemons  sur  les  machines  thermiques:  Moteurs  d  gas  et  d  petrolet 
given  at  Bordeaux  University  in  1899-1900. 


CHEMICAL  DYNAMICS  AND  EXPLOSIONS.  433 

detonation  under  the  actioa  of  the  various  agencies  we  have  just 
enumerated. 

Berthelot  and  Vielle  have  shown  that,  in  order  for  acetylene  to 
explode  in  contact  with  a  platinum  wire  brought  to  incandescence, 
it  was  necessary  to  submit  the  gas  to  an  initial  pressure  measured 
by  137  centimetres  of  mercury.  But  acetylene  detonates  by  the 
explosion  of  a  cartridge  containing  0.1  gr.  of  mercury  fulminate,  as 
soon  as  the  initial  pressure  is  measured  by  100  centimetres  of 
mercury. 

We  shall  cease  here  these  observations  on  chemical  dynamics 
and  explosions. 

While  the  chemical  statics  of  veritable  equilibria  is  already  so 
well  advanced  that  it  furnishes  a  great  number  of  exact  theorems, 
and  finds  numerous,  varied  and  exact,  experimental  confirmations, 
the  chemical  statics  of  false  equilibria  and  chemical  dynamics  are 
still  in  an  undeveloped  state.  Already,  however,  they  allow  of 
classifying  the  greater  number  of  the  reactions  observed  when  the 
temperature  and  pressure  in  a  chemical  system  are  varied;  until 
recently,  this  study  had  remained  a  veritable  chaos. 


LIST  OF  AUTHORS  CITED  IN  THIS  WORK. 


Adriani,  294,  295 
Alexyew,  314,  318 
Ampere,  331 
Andrews,  314,  324 
Arnold,  310 
Avogadro,  51,  331,  352 

B 

Bakhuis  Roozboom,  x,  113,  127,  152, 
159, 192, 193,  224,  225,  243,  251,  252, 
256,  265,  269,  271,  306,  311 

Berard,  49-52 

Berthelot,  40,  41,  42,  49,  51,  55,  99, 
210,  346,  417,  431,  432 

Berthollet,  40,  55,  204 

Bertrand,  333,  334 

Berzelius,  57 

Bineau,  223,  356 

Blagden,  204 

Bonnefoi,  179 

Bodenstein,  371 

Boulouch,  284 

Boyle,  356 

Braun,  200,  202 

Brown,  230 

Bruni,  273,  288,  289,  325 

Bunsen,  175 

Byl,  305 

C 

Cady,  289 

Cahours,  347,  348,  351,  356 

Cailletet,  324 

Cannizaro,  347 

Carnot  (Sadi),  35,  67,  75,  80,  85,  173 

Caubet,  324 

Caweth,  125 

Centnerszwer,  294 

Chancel,  221 

Chapeyron,  171*  175,  179 

Chappuis,  432 


Charles,  27,  356 

Charpy,  125,  298,  300,  301,  332 

Clausius,  26,  33,  67,  75,  80,  85,  86,  87, 

93,  171,  175 
Coppet  (de),  249 
Coulier,  164 
Crafts,  352,  354 
Croizier,  151 


Debray,  x,  57,  62,  65,  66,  72, 112, 113, 

151,  154,  178 
Delaroche,  49,  52 
Demerliac,  175 
De  venter  (van),  152,  296 
Ditte,  199,  211,  381,  408,  409 
Dittmar,  233,  238,  239 
Donnau,  133 
Donny,  366 
Duclaux,  325 
Dufour,  366 

Dupre-  (Athanase),  332,  334,  343 
Duhem,  372 


Engel,  376 

Estreicher-Rozbierski,  140 
Euler-Chelpin,  140 
Eyk  (van),  281,  282 


Fabre,  212,  346 

Favre,  ix,  41,  42,  179 

Fischer,  184 

Fock,  291 

Friedel  (Georges),  157 


Galileo,  402 
Garelli,  289 
Gauss,  363 

Gautier  (Armand),  386,  421,  429 
435 


436 


LIST  OF  AUTHORS  CITED  IN  THIS  WORK. 


Gautier  (Henri),  300 

Gay-Lussac,  27,  28 

Gernez,  x,  183,  366 

Gibbs  (J.  Willard),  ix,  88,  94,  106,  109, 
118,  223,  224,  227,  235,  263,  295, 
328,  329,  348,  350,  355,  356,  360 

Goldschmidt,  296 

Grenet,  250,  251,  259,  311 

Grotrian,  164 

Grove,  57 

Guldberg,  335-341 

Guthrie,  224,  232,  249 

H 

Hautefeuille,  x,  112,  113,  150,  158, 
184,  188,  209,  212,  324,  334,  348, 
350,  352,  390,  409,  410,  411,  432 

Helier,  386,  387,  388,  421,  429 

Helmholtz  (Herman  von),  94 

Helmholtz  (Robert  von),  184 

Heycock,  303 

Hissink,  276,  278,  280,  281,  303 

Hoitsema,  159 

Horstmann,  x,  88,  328,  329,  339,  356 


Isambert,  67,  113,  151,  155,  157,  335, 
339,  340,  341 


Joannis,  72,  151,  154,  335 
Jorissen,  296 
Joubert,  396 
Joule,  23,  28 
Jouniaux,  346,  374,  379 

K 

Kepler,  402 

Kirchhoff,  183,  184 

Konovalow,  147,  227,  229,  230,  295 

Kuenen,  324 

Kuriloff,  227 

Kuster,  289 


Laplace,  35,  36 

Lavoisier,  36,  387 

Le  Chatelier,  34,  113,  150,  152,  155, 

176,  202,  221,  225,  226,  259,  261, 

301,  306,  311,  421,  430 
Lehmann,  269 
Lemoine,  186,  198 
Levol,  303 
Loewel,  243 
Loewenberg,  133 
Lussana  (Silvio),  176 


M 

Malaguti,  55 

Mallard,  34,  150,  176,  421,  430 

Marchis,  432 

Mariotte,  273 

Massieu,  92 

Matignon,  41 

Maxwell  (J.  Clerk),  93 

Mayer  (Robert),  23,  31,  34,  37,  51 

Meier,  352,  355 

Meyer  (Victor),  429 

Meyerhoffer,  128,  133,   138,  140,  225, 

254 
Mitscherlich,  265,  288,  289,  348,  356, 

421 

Mohr,  269 
Morren,  394 
Moutier,  x,   117,   163,   164,   165,   166, 

167,  172,  183,  184,  185,  212 
Miiller,  297 
Miiller  (R.),  356 

N 

Natanson  (E.  &  L.),  356 
Naumann,  339,  356 
Neville,  303 
Newton,  361 

O 
Osmond,  303,  306,  310 

P 

Parmentier,  221 
Pasteur,  290-325 
Panchon,  220 

Pean  de  Saint-Giles,  54,  55 
Pelabon,  153,  154,  199,  369,  370,  371, 

380,  398,  401,  404,  408,  410 
Perrot,  206 
Peslin,  x,  178 
Pictet,  393 
Pickering,  225 
Playfair,  356 
Ponsot,  250 

R 

Ramsay,  184 

Raoult,  204 

Regnault  (Victor),  27,  28,  34,  182,  333 

Reicher,  152 

Reinders,  274,  275,  302,  303 

Retgers,  269,  270,  271,  326 

Riecke,  334 

Roberts-Austen,  299,  301,  303,  306 

Robin,  162,  164,  167,  172 

Robinson,  347 


LIST  OF  AUTHORS  CITED  IN  THIS  WORK. 


437 


Roland-Gosselin,  300,  302 
Romanow,  318 
Roscoe,  233,  238,  239 
Rothmund,  314,  318 
Riidorff,  262,  263 


Sainte-Claire  Deville  (Henri),  x,  35, 
57,  59,  60,  61,  65,  112,  113,  166,  206, 
208,  209,  387,  388,  389,  394,  402,  410 

Sarrau,  105 

Saunders,  138 

Schreinemakers,  127,  318,  325 

Silbermann,  41,  42,  179 

Snell,  325 

Sorby,  310 

Soret,  353 

Spring  (W.),  153,  318 

Stortenbeker,  226,  258,  265,  266,  267, 
269 

Stransfield,  301,  306 


Tammann,  157 

Thomsen,  99,  101,  166 

Thomson  (James),  150,  173,  175,  183 

Thomson  (William),  Lord  Kelvin,  28, 

150,  175 
Thorpe,  230 


Troost,  x,  112,  113,  150,  158,  184,  188 
209,  230,  334,  348,  352,  356,  390, 
409,  410 


Van  de  Stadt,  398 

Van  der  Heide,  127 

Van  der  Lee,  319 

Van  der  Waals,  ix,  324 

Van't  Hoff,  x,  128,  133,  137,  140,  142, 

143,  144,  202,  213,  225,  254,  264, 

296,  297,  429 
Van  Wyk,  253 
Vieille,  432 
Villard,  113,  152 
Visser  de,  175 

W 

Waage,  335,  341 
Wanklyn,  347,  356 
Wiedemann  (Gustav),  112 
Witkovski,  34 
Wroblewski,  113,  152 
Wiilner,  164 
Wiirtz,  348 


Young  (Sidney),  184 


INDEX  OF  CHEMICAL  SUBSTANCES 


STUDIED  IN  THIS  WORK 

Numbers  refer  to  pages. 


Acetic  acid,  175,  183,  184,  204,  230, 
325,  351,  356 

ether,  211,  230 
Acetylacetone,  318 
Acetylene,  43,  210,  433 
Air,  34,  324 
Alcohol,  amyl,  325 

butyl,  230,  318 

ethyl,  54,  230,  325,  334 

isobutyl,  318 

methyl,  229,  230,  318,  334 

propyl,  325 
Alum,  ammoniacal,  250 

potassic,  201 
Aluminium,  300 
Amines,  hydrates  of,  225 
Aminosuccinic  acid,  294 
Ammonia,  156,  334,  393,  394 
Ammonium,  bisulphide,  340 

carbonate,  336 

chloride,  200,  262,  269 

cyanide,  339 

nitrate,  105,  175 

picrate,  105 

racemate,  296 

sulphate,  262 

tartrate,  296  , 

Analcime,  157 
Aniline,  318 
Anthracene,  289 

Antimony,  298,  299,  301,  302,  303 
Arsenic,  336 

Arsenide  of  hydrogen,  432 
Astrakanite,  143,  146,  193 
Azobenzol,  288 


Barium,  chloride,  261 

nitrate,  250,  260 
Benzine,  175,  183,  204,  318 
Benzoltetrahydroquinaldine,  294 
Benzoic,  acid,  54,  318 

ether,  54 

Benzyl  aminosuccinic  acid,  294 
Bihydrocarvoxime,  296 
Bismuth,  125,  299,  300,  302,  318 
Blende,  hexagonal,  410 
Bromhydric  acid,  225 
Butyric  acid,  230 


Cadmium,  299 

amalgam,  305 

sulphate,  265,  269 
Calcite,  326 
Calcium,  acetate,  152,  194 

carbonate,  65,  178,  271,  326 

chloride,  225,  256 

isobutyrate  221 

nitrate,  260 

orthobutyrate,  220 

sulphate,  143,  145,  146,  261 
Carbazol,  289 
Carbon,  oxide,  30,  41,  51,  208 

sulphide,  230,  318,  334 

tetrachloride,  230,  334 
Carbonic,  anhydride,  41,  59,  312,  323, 

334,  388 

Carnallite,  129,  138,  143,  146 
Carvoxime,  295 
Cementite,  309,  311 
Chloracetic,  acid  mono-,  289 

439 


440 


INDEX  OF  CHEMICAL  SUBSTANCES. 


Chlorhydric   acid,  51,    127,   209,   225, 
393 

ether,  334 
Chlorides,  ammoniacal,  67,  150,   157, 

334 

Chlorine  hydrate,  155 
Chlorobenzine,  318 
Chloroform,  334 
Cobalt  chloride,  269 
f)  Collidine,  318 
Collodion,  105 
Copper,  298,  300,  302,  303 

acetate,  152,  194 

oxides,  154 

sulphate,  265,  266,  268,  269 
Cotton,  gun,  105 
Cyamelide,  411 
Cyanic  acid,  150,  411 
Cyanogen,  150 
Cyanuric  acid,  150,  411 

D 

Diacetyltartric,  dimethyl  ether  of  the 

acid,  294 

Diethylamine,  318 
Dolomite,  271 

E 

Essence  of  mustard,  318 
Ether,  324,  334 
Ethylamine,  hydrate,  225 
Ethyl  acetate,  211,  230 
Ethylene  dibromide,  204 


Ferrite,  311 

Firedamp,  430 

Formic  acid,  204,  230,  351,  356 

Furfurol,  318 

G 

Glaserite,  139,  143,  146 
Glauber  salts,  139 
Glauberite,  145,  146 
Gold,  301,  303 
Graphite,  307,  310 
Gypsum,  145,  146 

H 

Hexane,  318 

Hydrocarbide  of  bromine,  183 

Hydrogen,  30,  51,  61 


Iodine,  352,  354 

perchloride,  226,  258 
protochloride,  226,  258 


lodohydric  acid,  198,  346 
Iron,  61 

«,  A  r,  310 

carbides,  306 

casting,  308 

magnetic  oxide,  61 

perchloride,  127,  225,  251,  253,  269 

sulphate,  265,  269 
Isobutyric  acid,  318 


K 


Kainite,  143,  146 


Lead,  125,  298,  299,  300 

nitrate,  260 

Leonite,  128,  138,  143,  146 
Levol  alloy,  303 
Lithium,  ammoniacal  bromide,  179 

ammoniacal  chloride,  179 

borate,  225 

carbonate,  226,  261 

nitrate,  126 

M 

Magnesite,  326 

Magnesium,  carbonate,  271,  326,  376 

chloride,    129,    133,    137,    140,    142, 
143,  146,  225,  254 

platinocyanide,  157 

sulphate,    127,   133,  137,  142,  146, 

193,  265,  269 
Manganese,  chloride,  269 

sulphate,  265,  268,  269 
Martensite,  307 
Mercury,  334 

fulminate,  105 

iodide,  275,  277 

oxide,  153,  342 

sulphide,  337 
Methane,  430 
Methylbenzoic  ether,  294 
Methylethylacetone,  318 

N 

Naphthaline,  289 
/?  Naphthol,  227 
a  Naphthylamine,  175 
Nitric,  acid,  329 

oxide,  43,  51 

peroxide,  351,  355,  356 
Nitro-benzene,  204 
Nitrogen,  30,  51 
Nitroglycerine,  105 
Nitrous'  acid,  367 

oxide,  30,  41,  334 


INDEX  OF  CHEMICAL  SUBSTANCES. 


441 


O 
Oxime,  benzole,  295 

camphoric,  296 
Oxygen,  30,  51 
Ozone,  208,  353,  369,  389,  432 


Palladium,  hydride,  158 
Paracyanogen,  151 
Paraffine,  175 
Paratoluidine,  175 
Perchloric  acid  hydrates,  253 
Pertite,  310 
Phenanthrene,  289 
Phenylammonium,  phenolate  of,  318 
Phenylglycolic  acid,  294 
Phenol,  314,  318,  319 
Phosphide  of  hydrogen,  398 
Phosphonium,  bromide  of,  340 
Phosphorus,  184,  284,  334,  396,  410 

perchloride,  346 
Picric  acid,  105,  227 
Pinonic  acid,  292 
Platinum  chlorides,  409 
Potash,  393 

Potassammonium,  152,  335 
Potassium,  acetate,  56 

bicarbonate,  376 

carbonate,  226 

chlorate,  250 

chloride,    128,    133,   138,   139,   140, 
142,  143,  146,  250 

chromate,  288 

nitrate,  56,  125,  250,  260,  278,  282 

sulphate,  127,   133,   138,   139,   250, 

260,  262 

Powder,  gun,  105 
Propionitrite,  318 


Racemic  acid,  290 
Resorcine,  313 
Rubidium,  racemate,  297 
tartrate,  297 

S 

Salicylic  acid,  318 
Scho^nite,  128,  138,  143 
Selenhydric  acid,   199,  211,  344,  379, 

403,  406 
Selenium,  285 
Silicon  chlorides,  390,  409 

fluorides,  409 
Silver,  302,  303 


Silver,  ammoniacal  bromide,  335 

ammoniacal  cyanide,  335 

ammoniacal  iodide,  335 

ammoniacal  nitrate,  335 

bromide,  347 

chloride,  345,  374,  379 

iodide,  175 

nitrate,  276,  281,  305 
Soda,  393,  394 
Sodammonium,  151,  335 
Sodium,  borate,  226 

carbonate,  261 

chloride,  138,  140,  143,  249,  261 

nitrate,  125,  260,  276,  278,  281,  302 

pyrophosphate,  226,  261 

racemate,  297 

sulphate,   138,   140,   142,   143,   146, 
193,  201,  220,  243,  250 

tartrate,  297 
Spermaceti,  175 
Stilbene,  288 
Strontium,  acetate,  56 

nitrate,  56,  260 
Succinonitrite,  318 
Sulphydric  acid,  334,  371,  375,  378, 

Sulphur,  284,  285,  334,  350,  357,  367, 

Sulphuric  acid,  225,  393,  394 
Sulphurous  anhydride,  208 
Syngenite,  145,  146 

T 

Tartaric  acid,  290,  294 
Tellurium,  409 
Thallium  nitrate,  282 
Thorium,  sulphate,  243 
Tin,  125,  298,  299,  300,  301 
Toluene,  318 
Tournesol,  394 
Trie  thy  lamine,  318 
Turpentine,  334 

W 

Water,  46,  49,  57,  58,  61, 175, 183, 184, 

185,  204,  318,  325,  333,  334 
Wiirtzite,  409 

Z 

Zeolites,  157 

Zinc,  298,  299,  300,  303, 318 

alloys,  303 

oxide,  409 

sulphate,  265,  266 

sulphide,  409 


GENERAL  INDEX. 


Numbers  refer  to  pages. 


Adiabatic,  101 

Allotropic  transformations,  150,  175 

Alloys,  125,  298  et  seq. 

Amalgams,  305 

Attraction  hypotheses,  361 

Axes,  co-ordinate,  7 

Bi VARIANT  SYSTEMS,  114,  214  et  seq. 
Bomb,  calorimetric,  41 

CALORIMETRY,  chemical,  36  et  seq. 
Capillary  actions,  357  et  seq. 
Carburized  iron,  306  et  seq. 
Carnot,  principle  of,  75,^80,  85 
C.  G.  S.  system,  1 
Clapeyron,  law  of,  171 
Clausius,  law  of,  34,  171 

form  of  second  law,  25 

principle  of,  76,  80,  85 
Combination,  region  of,  373 
Components,  independent,  106,  150 
Compound,  existence  of,  156,  158 
Concentrations,  of  bivariant    systems, 

115 
Condensation,  normal  and  retrograde, 

322 
Critical  states,  312  et  seq. 

point,  312 

temperature,  313,  316 

temperature  and  pressure,  319 

volume,  313 

line,  319 

Cryohydrates,  249 
Crystals,  mixed,  262  et  seq.,  288,  292 

limiting  forms,  325 
Cycle,  closed,  23 

isothermal,  78 

open,  24 

DECOMPOSITION,  region  of,  378 
Delaroche  and  Board's  law,  49 
Dew  surface,  319;  line  321 ;  point,  322 


Displacements,  virtual,  17 

for  conservative  systems,  19 

of  equilibrium,  195  et  seq. 
Dissociation,  57,  59,  65, 178, 205,  etc. 

and  temperature  influence,  342 

domain  of,  387 

in  vacuo,  349 

minimum,  211 

of  polymers,  351 

tension  of,  65, 112, 150, 158, 334 
Distillation,  230 

of  mixtures,  236 

temperature  and  pressure  effects,  238 
Domain,  of  a  precipitate,  126, 132 
Dynamics,  chemical,  412  et  seq. 

EBULLITION  SURFACE,  319,  321 

Electric  spark,  206 

Endothermic  reactions,  41,   102.  205 

389  et  seq. 
Energy,  available,  93 

conservation  of,  18,  26 

free,  94 

internal,  25, 180 ;  of  a  gas,  29 

kinetic,  16 

potential,  26 

theorem  of,  16 

total,  26,  70 
Entropy,  81,  88,  93 

of  a  gas,  82 
Equilibrium,  and  pressure,  196 

and  temperature,  202 

apparent  false,  357  et  seq. 

chemical,  53 

displacement,  195  et  seq. 

false,  117,  164,  190,  369  et  seq.;  and 
friction,  398;  and  stability,  421 

indifferent,  171, 196, 223, 423 

isothermal,  90,  94 

law  for  gases,  331 

limiting,  421,  et  seq. 


444 


GENERAL  INDEX. 


Equilibrium  of  bi variant  systems,  114, 
214 

physical  states  of,  69,  72 

stability  of,  19,  96,  196 
Equivalent,  mechanical,  22,  23,  35 
Equivalence,  of  heat  and  work,  22,  24 

25,  76 

Etherification,  211,  416 
Eutectic  mixture,  247,  280 

point,  251,  278,  281 
Eutexia,  240  et  seq. 
Exothermic   reactions,  41,   102,   205, 

349 

Explosion-point,  429,  431 
Explosions,  412  et  seq. 
Explosives,  103,  430 

FORMATION,  heats  of,  44 

Forces,  internal,  95 

Friction  vs.  false  equilibrium,  398 

Function,  characteristic,  91,  93;  of  a 

gas,  92 
Fusion,  149 

aqueous,  223 

and  Clausius'  law,  173 

GAS,  and  solid  polymer,  150 

expansion  in  vacuo,  28 

heat  of  formation,  46,  49 

internal  energy  of,  29,  30 

laws  of,  11,  27,  29 

perfect,  30,  37,  327  et  seq. 

specific  heat  of,  30,  31 
Gay-Lussac's  experiment,  28 
Geometrical  representation,  7 
Grove's  experiment,  57 
Guldberg  and  Waage's  law,  335 

HEAT  and  initial  and  final  states,  36,  38 

engine,  80 

from  a  machine,  21 

of  formation,  44 

of  reaction,  321 

mechanical  equivalent  of,  22,  35 
Hydrate,  solubility  curves  of,  222,  240 

INDIFFERENT  EQUILIBRIUM,  223 

point,  214,  223,  227,  314 
Invariant  system,  110 
Isomorphous  mixtures,  262  et  seq.,  288 
Isodimorphous  mixtures,  275 
Isotetramorphous  mixtures,  281 
Isotrimorphous  mixtures,  281 
Isothermal  cycle,  78 

equilibrium,  90,  94 

reaction,  415 

transformation,  76,  88,  99,  102 


MACHINE,  continuously  acting,  80 

properties  of,  21 
Mariotte,  law  of,  11,  27 
R.  Mayer's  equation,  33 
Mechanics,  chemical,  67,  106 

of  perfect  gases,  327  et  seq. 
Mineralization,  410 
Mixtures,  distillation  of,  232,  236 

double,  214,  227,  314 

eutectic,  247 

isomorphous,  262 

non-explosive,  430 

of  crystals,  262  et  seq. 

of  melted  alloys,  232 

of  salts,  259 

of  volatile  liquids,  235 
Monovariant  system,  110,  113,  147  et 

seq. 

Motion,  perpetual,  79 
Motor,  80 

Moutier's  law,  163,  165;  corollary,  165 
Multiple  points,  180 
Multi variant  systems,  118  et  seq. 

OPTICAL  ANTIPODES,  288  et  seq. 

PHASE,  108 

Phase  rule,  106  et  seq. ;  contradictions 

to,  113 

Polymer,  solid,  150 
Potential,  14,  18 

internal    thermodynamic,    94,    109, 
359 

of  explosion,  104 

of  a  gas,  329 

Precipitate,  domain  of,  126 
Pressure,  work  due  to,  5 

vapor,  63 

QUADRIVARIANT  SYSTEMS,  130 

Quadruple  point,  192 
Quiptivariant  system,  140 
Quintuple  point,  193 

RACEMIC  COMPOUNDS,  291  et  seq. 
Reaction,  acceleration  of,  413,  416,  420 

adiabatic,  420,  424 

explosive,  420  et  seq. 

intensity,  98 

inverse,  56 

isothermal,  415,  418 

limited,  54,  59,  391 

moderated,  415 

point,  394,  429 

unlimited,  53,  384,  391 

velocity,  410,  417;  and  composition, 
415;  and  temperature,  416 


(.'I-: \ERAL  INDEX. 


445 


Reversible  transformation,  67,  70,  73, 

80,  378, 402 
Robin's  law,  162 

SALTS  (theory  of),  double,  118 

mixtures  of,  259 

solubility  surface,  120 

transformation  point,  152 
Saponification,  54 
Saturation,  of  solution,  63 
Solubility,  and  pressure,  200 

curve,  216,  222,  240 

of  isomorphous  salts,  264 

surface  of,  120,  131 
Solution,  concentration  of,  204,  216 

freezing-point  lowering,  203 

saturation,  63,  196,  216 

solid,  156,  263,  301 

supersaturated,  86 

unsaturated,  217 

vapor  tension  of,  204 
Specific  heat,  of  gases,  30-33 
Surfusion,  164,  185 

TEMPERATURE,  absolute,  27,  77 
Tension,  of  dissociation,  65,  112,  150, 

158,  334 
of  transformation,    111,    148,    186; 

curve  of,  149,  153,  167 
of  vaporization,  63,  111,  332 
Transformation,  compensated  and  non- 
compensated,  86 
isothermal,  76 


Transformation  point,  111,  149,152, 181 

renver sable,  68,  80 

reversible,  67,  72 

temperature,  112 

tension,  111,  148,  153,  167,  186 

value  of,  76,  81 
Transition,  and  eutexia,  240  et  seq. 

points  180,  189, 192, 241, 276 

temperature,  192,  276 
Triple  point,  181,  189 
Tri variant  systems,  118 

UNEQUAJ^LY  heated  spaces,  403  et  seq 

VAPOR,  of  water  action  on  iron,  61 

density,  351, 354 

tension,  63,  149,  332 
Vaporization,  319,  410 

and  reversibility,  70 

and  Clausius'  law,  173 

and  critical  point,  312 
Variance,  109 

and  equilibrium,  196 
Volatile  liquids,  228 

mixtures,  235 
Volatilization,  apparent,  409 

WATT'S  principle,  62,  410 
Weight,  work  of,  3,  5 
Work,  1  et  seq.,  88 

maximum,  99,  166,  212,  391 
compensated,  91,  97 


SHORT-TITLE     CATALOGUE 

OP  THE 

PUBLICATIONS 


OF 


JOHN   WILEY   &    SONS, 

NEW  YORK, 
LOKDON:   CHAPMAN  &  HALL,  LIMITED. 

ARRANGED  UNDER  SUBJECTS. 


Descriptive  circulars  sent  on  application.  Books  marked  with  an  asterisk  (*)  are  sold 
at  net  prices  only,  a  double  asterisk  (**)  books  sold  under  the  rules  of  the  American 
Publishers'  Association  at  net  prices  subject  to  an  extra  charge  for  postage.  All  book: 
are  bound  in  cloth  unless  otherwise  stated. 


AGRICULTURE. 

Armsby's  Manual  of  Cattle-feeding i2mo,  Si  75 

Principles  of  Animal  Nutrition 8vo,  4  oo 

Budd  and  Hansen's  American  Horticultural  Manual: 

Part  I.  Propagation,  Culture,  and  Improvement. i2mo,  i  50 

Part  IL  Systematic  Pomology i2mo,  i  50 

Downing's  Fruits  and  Fruit-trees  of  America 8vo,  5  oo 

Elliott's  Engineering  for  Land  Drainage , i2mo,  i  50 

Practical  Farm  Drainage i2mo,  i  oo 

Green's  Principles  of  American  Forestry I2mo,  i  50 

Grotenfelt's  Principles  of  Modern  Dairy  Practice.     (WolL) i2mo,  2  oo 

Kemp's  Landscape  Gardening i2mo,  2  50 

Maynard's  Landscape  Gardening  as  Applied  to  Home  Decoration lamo,  i  *o 

Sanderson's  Insects  Injurious  to  Staple  Crops I2mo,  i  50 

Insects  Injurious  to  Garden  Crops.     (In  preparation.) 
Insects  Injuring  Fruits.     (In  preparation.) 

Stockbridge's  Rocks  and  Soils 8vo,  2  50 

Woll's  Handbook  for  Farmers  and  Dairymen i6mo,  i  50 

ARCHITECTURE. 

Baldwin's  Steam  Heating  for  Buildings :i2mo,  2  50 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  5  oo 

Birkmire's  Planning  and  Construction  of  American  Theatres 8vo,  3  oo 

Architectural  Iron  and  SteeL 8vo,  3  50 

Compound  Riveted  Girders  as  Applied  in  Buildings 8vo,  2  oo 

Planning  and  Construction  of  High  Office  Buildings 8vo  3  50 

Skeleton  Construction  in  Buildings 8vo,  3  oo 

Brigg's  Modern  American  School  Buildings 8vo,  4  oo 

Carpenter's  Heating  and  Ventilating  of  Buildings 8vo,  4  oo 

Freitag's  Architectural  Engineering 8vo,  3  50 

Fireproofing  of  Steel  Buildings 8vo,  2  50 

French  and  Ives's  Stereotomy 8vo,  2  50 

Gerhard's  Guid«  to  Sanitary  House-inspection i6mo,  i  oo 

Theatre  Fires  and  Panics I2mo,  i  50 

Holly's  Carpenters'  and  Joiners'  Handbook i8mo,  75 

Johnson's  Statics  by  Algebraic  and  Graphic  Methods 8vo,  2  oo 


Kidder's  Architects' and  Builders' Pocket-book.  Rewritten  Edition.  i6mo,mor.f  5  oo 

Merrill's  Stones  for  Building  and  Decoration 8vo,  5  oo 

Non-metallic  Minerals:   Their  Occurrence  and  Uses 8vo,  4  oo 

Monckton's  Stair-building 4to,  4  oo 

Patton's  Practical  Treatise  on  Foundations 8vo,  5  oo 

Peabody's  Naval  Architecture Svo,  7  50 

Richey's  Handbook  for  Superintendents  of  Construction i6mo,  mor  ,  4  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish Svo,  3  oo 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry Svo,  i  50 

Snow's  Principal  Species  of  Wood Svo,  3  50 

Sondericker's  Graphic  Statics  with  Applications  to  Trusses,  Beams,  and  Arches. 

Svo,  2     3 

Towne's  Locks  and  Builders'  Hardware iSmo,  morocco,  3  oo 

Wait's  Engineering  and  Architectural  Jurisprudence Svo,  6  oo 

Sheep,  6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  Svo,  5  oo 

Sheep,  5  50 

Law  of  Contracts Svo,  3  oo 

Wood's  Rustless  Coatings:   Corrosion  and  Electrolysis  of  Iron  and  Steel.  .Svo,  4  oo 

Woodbury's  Fire  Protection  of  Mills Svo,  2  50 

Worcester  and  Atkinson's  Small  Hospitals,  Establishment  and  Maintenance, 
Suggestions  for  Hospital  Architecture,  with  Plans  for  a  Small  Hospital. 

121110,  I     25 

The  World's  Columbian  Exposition  of  1893 Large  4to,  i  oo 

ARMY  AND  NAVY. 

Bernadou's  Smokeless  Powder,  Nitro-cellulose,  and  the  Theory  of  the  Cellulose 

Molecule i2mo,  2  50 

*  Bruff 's  Text-book  Ordnance  and  Gunnery Svo,  6  oo 

Chase's  Screw  Propellers  and  Marine  Propulsion Svo,  3  oo 

Cloke's  Gunner's  Examiner.     (In  press.) 

Craig's  Azimuth 4to,  3  50 

Crehore  and  Squier's  Polarizing  Photo-chronograph Svo,  3  oo 

Cronkhite's  Gunnery  for  Non-commissioned  Officers 24010,  morocco,  2  oo 

*  Davis's  Elements  of  Law Svo,  2  50 

*  Treatise  on  the  Military  Law  of  United  States Svo,  7  oo 

Sheep,  7  50 

De  Brack's  Cavalry  Outposts  Duties.     (Carr.) 24mo,  morocco,  2  oo 

Dietz's  Soldier's  First  Aid  Handbook i6mo,  morocco,  i  25 

*  Dredge's  Modern  French  Artillery 4to,  half  morocco,  15  oo 

Durand's  Resistance  and  Propulsion  of  Ships 8vo,  5  oo 

*  Dyer's  Handbook  of  Light  Artillery I2mo,  3  oo 

Eissler's  Modern  High  Explosives Svo,  4  oo 

*  Fiebeger's  Text-book  on  Field  Fortification Small  Svo,  2  oo 

Hamilton's  The  Gunner's  Catechism i8mo,  i  oo 

*  Hoff' s  Elementary  Naval  Tactics Svo,  i  50 

Ingalls's  Handbook"  of  Problems  in  Direct  Fire Svo,  4  oo 

*  Ballistic  Tables Svo,  i  50 

*  Lyons's  Treatise  on  Electromagnetic  Phenomena.  Vols.  I.  and  II. .  Svo,  each,  6  oo 

*  Mahan's  Permanent  Fortifications.    (Mercur.) Svo,  half  morocco,  7  50 

Manual  for  Courts-martial i6mo,  morocco,  i  30 

*  Mercur's  Attack  of  Fortified  Places i2mo,  2  oo 

*  Elements  of  the  Art  of  War Svo,  4  oo 

Metcalf's  Cost  of  Manufactures — And  the  Administration  of  Workshops.  .Svo,  5  oo 

*  Ordnance  and  Gunnery.     2  vols i2mo,  5  oo 

Murray's  Infantry  Drill  Regulations iSmo,  paper,  10 

Ifixon's  Adjutants'  Manual 24010,  i  oo 

Peabody's  Naval  Architecture Svo,  7  50 

3 


*  rhelps's  Practical  Marine  Surveying 8vo,  2  50 

Powell's  Army  Officer's  Examiner i2mo,  4  oo 

Sharpe's  Art  of  Subsisting  Armies  in  War i8mo.  morocco,  i  50 

*  Walke's  Lectures  on  Explosives 8vo,  4  oo 

*  Wheeler's  Siege  Operations  and  Military  Mining 8vo,  2  oo 

Winthrop's  Abridgment  of  Military  Law i2mo,  2  50 

WoodhulTs  Notes  on  Military  Hygiene i6mo,  i  50 

Young's  Simple  Elements  of  Navigation i6mo,  morocco,  i  oo 

Second  Edition,  Enlarged  and  Revised i6mo,  morocco,  2  oo 

ASSAYING. 

Fletcher's  Practical  Instructions  it.  Quantitative  Assaying  with  the  Blowpipe. 

I2mo,  morocco,  i  50 

Furman's  Manual  of  Practical  Assaying 8vo,  3  oo 

Lodge's  Notes  on  Assaying  and  Metallurgical  Laboratory  Experiments ....  8vo,  3  oo 

Miller's  Manual  of  Assaying i2mo,  i  oo 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  2  oo 

Ricketts  and  Miller's  Notes  on  Assaying 8vo,  3  oo 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  oo 

Wilson's  Cyanide  Processes i2mo,  i  50 

Chlorination  Process -. i2mo,  i  50 

ASTRONOMY. 

Comstock's  Field  Astronomy  for  Engineers 8vo,  2  50 

Craig's  Azimuth 4to,  3  50 

Doolittle's  Treatise  on  Practical  Astronomy 8vo,  4  oo 

Gore's  Elements  of  Geodesy 8vo,  2  50 

Hayford's  Text-book  of  Geodetic  Astronomy 8vo,  3  oo 

Merriman's  Elements  of  Precise  Surveying  and  Geodesy 8vo,  2  50 

*  Michie  and  Harlow's  Practical  Astronomy 8vo,  3  oo 

*  White's  Elements  of  Theoretical  and  Descriptive  Astronomy i2mo,  2  oo 

BOTANY. 

Davenport's  Statistical  Methods,  with  Special  Reference  to  Biological  Variation. 

i6mo,  morocco,  i  25 

Thome'  and  Bennett's  Structural  and  Physiological  Botany i6mo,  2  25 

Westermaier's  Compendium  of  General  Botany.     (Schneider.) 8vo,  2  oo 

CHEMISTRY. 

Adriance's  Laboratory  Calculations  and  Specific  Gravity  Tables i2mo,  i  25 

Allen's  Tables  for  Iron  Analysis 8vo,  3  oo 

Arnold's  Compendium  of  Chemistry.     (Mandel.) Small  8vo,  3  50 

Austen's  Notes  for  Chemical  Students I2mo,  i  50 

Bernadou's  Smokeless  Powder. — Nitro-cellulose,  and  Theory  of  the  Cellulose 

Molecule i2mo,  2  50 

Bolton's  Quantitative  Analysis 8vo,  i  50 

*  Browning's  Introduction  to  the  Rarer  Elements 8vo,  i  50 

Brush  and  Penfield's  Manual  of  Determinative  Mineralogy. 8vo,  4  oo 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.    (Boltwood. ).  .8vo,  3  oo 

Cohn's  Indicators  and  Test-papers i2mo,  2  oo 

Tests  and  Reagents 8vo,  3  oo 

Crafts's  Short  Course  in  Qualitative  Chemical  Analysis.   (Schaeffer.). .  .i2mo,  i  50 
Dolezalek's  Theory  of  the  Lead  Accumulator  (Storage  Battery).        (Von 

Ende.) i2mo,  2  50 

Drechsel's  Chemical  Reactions.     (Merrill.) i2mo,  i  25 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.) 8vo,  4  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

Effront's  Enzymes  and  their  Applications.     (Prescott.) 8vo,  3  oo 

Erdmann's  Introduction  to  Chemical  Preparations.     (Dunlap.) i2mo,  i  35 

3 


Fletcher's  Practical  Instructions  in  Quantitative  Assaying  with  the  Blowpipe. 

i2mo,  morocco,  i  50 

Fowler's  Sewage  Works  Analyses i2mo,  2  oo 

Fresenius's  Manual  of  Qualitative  Chemical  Analysis.     (Wells.) 8vo,  5  oc 

Manual  of  Qualitative  Chemical  Analysis.  Part  I.  Descriptive.  (Wells.)  8vo,  3  o* 
System   of    Instruction   in    Quantitative    Chemical   Analysis.      (Cohn.) 

2  vols 8vo,  12  50 

Fuertes's  Water  and  Public  Health i2mo,  i  50 

Furman's  Manual  of  Practical  Assaying ' 8vo ,  3  oo 

*  Getman's  Exercises  in  Physical  Chemistry i2mo,  2  oc 

Gill's  Gas  and  Fuel  Analysis  for  Engineers i2mo,  i  25 

Grotenfelt's  Principles  of  Modern  Dairy  Practice.     (Woll.) i2mo,  2  oo 

Hammarsten's  Text-book  of  Physiological  Chemistry.     (Mandel.) 8vo,  4  oo 

Helm's  Principles  of  Mathematical  Chemistry.     (Morgan.) i2mo,  i  50 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Hind's  Inorganic  Chemistry 8vo,  3  oc 

*  Laboratory  Manual  for  Students i2mo,         75 

Holleman's  Text-book  of  Inorganic  Chemistry.     (Cooper.) 8vo,  2  50 

Text-book  of  Organic  Chemistry.     (Walker  and  Mott.) 8vo,  2  50 

*  Laboratory  Manual  of  Organic  Chemistry.     (Walker.) i2mo,  i  oo 

Hopkins's  Oil-chemists'  Handbook 8vo,  3  oo 

Jackson's  Directions  for  Laboratory  Work  in  Physiological  Chemistry.  .8vo,  i  25 

Keep's  Cast  Iron 8vo,  2  50 

La4d's  Manual  of  Quantitative  Chemical  Analysis i2mo,  i  oo 

Landauer's  Spectrum  Analysis.     (Tingle.) 8vo,  3  oo 

*  Langworthy   and   Austen.        The   Occurrence   of  Aluminium  in  Vege  able 

Products,  Animal  Products,  and  Natural  Waters 8vo,  2  oo 

Lassar-Cohn's  Practical  Urinary  Analysis.  (Lorenz.) I2mo,  i  oo 

Application  of  Some  General  Reactions  to  Investigations  in  Organic 

Chemistry.  (Tingle.) i2mo,  i  oo 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control. 8yo,  7  50 

Lob's  Electrolysis  and  Electrosynthesis  of  Organic  Compounds.  (Lorenz. ).i2mo,  i  oo 

Lodge's  Notes  on  Assaying  and  Metallurgical  Laboratory  Experiments. ..  .8vo,  3  oo 

Lunge's  Techno-chemical  Analysis.  (Cohn.) i2mo,  i  oo 

Mandel's  Handbook  for  Bio-chemical  Laboratory i2mo,  i  50 

*  Martin's  Laboratory  Guide  to  Qualitative  Analysis  with  the  Blowpipe .  .  i2mo,        60 
Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

3d  Edition,  Rewritten 8vo,  4  oo 

Examination  of  Water.     (Chemical  and  Bacteriological.) i2mo,  i  25 

Matthew's  The  Textile  Fibres 8vo,  3  50 

Meyer's  Determination  of  Radicles  in  Carbon  Compounds.     (Tingle.).  .  lamo,  i  oo 

Miller's  Manual  of  Assaying i2mo,  i  oo 

Mixter's  Elementary  Text-book  of  Chemistry i .  i2mo,  i  50 

Morgan's  Outline  of  Theory  of  Solution  and  its  Results i2ino,  i  oo 

Elements  of  Physical  Chemistry i2mo,  2  oo 

Morse's  Calculations  used  in  Cane-sugar  Factories. i6mo,  morocco,  i  50 

JVIulliken's  General  Methodt  for  the  Identification  of  Pure  Organic  Compounds. 

Vol.  I Large  8vo,  5  oo 

O'Brine's  Laboratory  Guide  in  Chemical  Analysis 8vo,  2  oo 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  2  oo 

Ostwald's  Conversations  on  Chemistry.     Part  One.     (Ramsey.) I2mo,  150 

Ostwald's  Conversations  on  Chemistry.     Part  Two.     (Turnbull.).     (In  Press.) 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Miceral  Tests. 

8vo,  paper,         50 

Pictet's  The  Alkaloids  and  their  Chemical  Constitution.     (Biddle.)  ....  .8vo,  5  oo 

Pinner's  Introduction  to  Organic  Chemistry.     (Austen.) ismo,  i  50 

Poole's  Calorific  Power  of  Fuels 8vo,  3  oo 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis i2mo,  i  25 

4 


*  Reisig's  Guide  to  Piece-dyeing.  .  .      8vo,  25  oo 

Richards  and  Woodman's  Air,  Water,  and  Food  from  a  Sanitary  Standpoint  8vo,  2  oo 

Richards's  Cost  of  Living  as  Modified  by  Sanitary  Science 12 mo,  i  oo 

Cost  of  Food,  a  Study  in  Dietaries i2mo,  i  oo 

*  Richards  and  Williams's  The  Dietary  Computer 8vo,  i  50 

Ricketts  and  Russell's  Skeleton  Notes  upon  inorganic  Chemistry.     (Part  I. 

Non-metallic  Elements.) 8vo,  morocco,  75 

Ricketts  and  Miller's  Notes  on  Assaying 8vo,  3  oo 

Rideal's  Sewage  and  the  Bacterial  Purificat'on  of  Sewage 8vo,  3  50 

Disinfection  and  the  Preservation  of  Food 8vo,  4  oo 

Rigg's  Elementary  Manual  for  the  Chemical  Laboratory 8vo,  i  25 

Rostoski's  Serum  Diagnosis.  (Bolduan.) i2mo,  i  oo 

Ruddiman's  Incompatibilities  in  Prescriptions 8vo,  2  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Salkowski's  Physiological  and  Pathological  Chemistry.  (Orndorff.) 8vo,  2  50 

Schimpf's  Text-book  of  Volumetric  Analysis i2mo,  2  50 

Essentials  of  Volumetric  Analysis.  .  , i2mo,  i  25 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco,  3  oo 

Handbook  for  Sugar  Manufacturers  and  their  Chemists.  .  i6mo,  morocco,  2  oo 

Stockbridge's  Rocks  and  Soils 8vo,  2  50 

*  TiUman's  Elementary  Lessons  in  Heat 8vo,  i  50 

*  Descriptive  General  Chemistry 8vo,  3  oo 

TreadwelTs  Qualitative  Analysis.     (HalL) 8vo.  3  oo 

Quantitative  Analysis.     (HalL) 8vo,  4  oo 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  oo 

Van  Deventer's  Physical  Chemistry  for  Beginners.     (Boltwood.) 12 mo,  i  50 

*  Walke's  Lectures  on  Explosives 8vo,  4  oo 

Washington's  Manual  of  the  Chemical  Analysis  of  Rocks 8vo,  2  oo 

Wassermann's  Immune  Sera :  Hsemolysins,  Cytotoxins,  and  Precipitins.    (Bol- 
duan.)   i2mo,  i  oo 

Well's  Laboratory  Guide  in  Qualitative  Chemical  Analysis ". 8vo,  i  50 

Short  Course  in  Inorganic  Qualitative  Chemical  Analysis  for  Engineering 

Students i2mo,  i  50 

Text-book  of  Chemical  Arithmetic.     (In  press.) 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Wilson's  Cyanide  Processes i2mo,  i  50 

Chlorination  Process i2mo,  i  50 

Wulling's    Elementary    Course    in  Inorganic,  Pharmaceutical,  and  Medical 

Chemistry i2mo,  2  oo 

CIVIL  ENGINEERING. 

BRIDGES   AND   ROOFS.       HYDRAULICS.       MATERIALS   OF    ENGINEERING. 
RAILWAY  ENGINEERING. 

Baker's  Engineers'  Surveying  Instruments i2mo,  3  oo 

Bixby's  Graphical  Computing  Table Paper  19^  X  24^  inches.  25 

**  Burr's  Ancient  and  Modern  Engineering  and  the  Isthmian  Canal.     (Postage, 

27  cents  additional) 8vo,  3  50 

Comstock's  Field  Astronomy  for  Engineers 8vo,  2  50 

Davis's  Elevation  and  Stadia  Tablet 8vo,  i  oo 

Elliott's  Engineering  for  Land  Drainage i2mo,  i  50 

Practical  Farm  Drainage i2mo,  i  oo 

Fiebeger's  Treatise  on  Civil  Engineering.     (In  press.) 

Folwell's  Sewerage.     (Designing  and  Maintenance.) 8vo,  3  oo 

Freitag's  Architectural  Engineering.     2d  Edition,  Rewritten 8vo,  3  50 

French  and  Ives's  Stereotomy 8vo,  2  50 

Goodhue's  Municipal  Improvements xarro,  i  75 

Goodrich's  Economic  Disposal  of  Towns'  Refuse 8vo,  3  50 

Gore's  Elements  of  Geodesy 8vo,  2  50 

Hayford's  Text-book  of  Geodetic  Astronomy &vo,  3  oo 

Hering's  Ready  Reference  Tables  (Conversion  Factors') i6mo,  morocco.  3  50 

5 


Howe's  Retaining  Walls  for  Earth i2mo,  i  25 

Johnson's  (J.  B.)  Theory  and  Practice  of  Surveying Small  8vo,  4  oo 

Johnson's  (L.  J.)  Statics  by  Algebraic  and  Graphic  Methods 8vo,  2  oo 

Laplace's  Philosophical  Essay  on  Probabilities.    (Truscott  and  Emory.)-  i2mo,  2  oo 

Mahan's  Treatise  on  Civil  Engineering.     (1873.)     (Wood.) 8vo,  5  oo 

*  Descriptive  Geometry 8vo,  i  50 

Merriman's  Elements  of  Precise  Surveying  and  Geodesy 8vo,  2  50 

Elements  of  Sanitary  Engineering 8vo,  2  oo 

Merriman  and  Brooks's  Handbook  for  Surveyors i6mo,  morocco,  2  oo 

Nugent's  Plane  Surveying 8vo,  3  50 

Ogden's  Sewer  Design i2mo,  2  oo 

Patton's  Treatise  on  Civil  Engineering 8vo  half  leather,  7  50 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  oo 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage 8vo,  3  50 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry t 8vo,  i  50 

Smith's  Manual  of  Topographical  Drawing.     (McMillan.) 8vo,  2  50 

Sondericker's  Graphic  Statics,  with  Applications  to  Trusses,  Beams,  and  Arches. 

8vo,  2  oo 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  oo 

*  Trautwine's  Civil  Engineer's  Pocket-book i6mo,  morocco,  5  oo 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  oo 

Sheep,  6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo,  5  oo 

Sheep,  5  50 

Law  of  Contracts 8vo,  3  oo 

Warren's  Stereotomy — Problems  in  Stone-cutting 8vo,  2  50 

Webb's  Problems  in  the  Use  and  Adjustment  of  Engineering  Instruments. 

i6mo,  morocco,  i  25 

*  Wheeler  s  Elementary  Course  of  Civil  Engineering 8vo,  4  oo 

Wilson's  Topographic  Surveying 8vo,  3  50 

BRIDGES  AND  ROOFS. 

Boiler's  Practical  Treatise  on  the  Construction  of  Iron  Highway  Bridges .  .  8vo,  2  oo 

*  Thames  River  Bridge 4to,  paper,  5  oo 

Burr's  Course  on  the  Stresses  in  Bridges  and  Roof  Trusses,  Arched  Ribs,  and 

Suspension  Bridges 8vo,  3  50 

Burr  and  Falk's  Influence  Lines  for  Bridge  and  Roof  Computations.  .  .  .8vo,  3  oo 

Du  Bois's  Mechanics  of  Engineering.     Vol.  II Small  4to,  10  oo 

Foster's  Treatise  on  Wooden  Trestle  Bridges 4to,  5  oo 

Fowler's  Ordinary  Foundations 8vo,  3  50 

Greene's  Roof  Trusses 8vo,  i  25 

Bridge  Trusses 8vo,  2  50 

Arches  in  Wood,  Iron,  and  Stone 8vo,  2  50 

Howe's  Treatise  on  Arches 8vo,  4  oo 

Design  of  Simple  Roof- trusses  in  Wood  and  Steel 8vo,  2  oo 

Johnson,  Bryan,  and  Turneaure's  Theory  and  Practice  in  the  Designing  of 

Modern  Framed  Structures Small  4to,  10  oo 

Merriman  and  Jacoby's  Text-book  on  Roofs  and  Bridges : 

Part  I.     Stresses  in  Simple  Trusses 8vo,  2  50 

Part  II.     Graphic  Statics 8vo,  2  50 

Part  in.     Bridge  Design 8vo,  2  50 

Part  IV.     Higher  Structures 8vo,  2  50 

Morison's  Memphis  Bridge 4to,  10  oo 

Waddell's  De  Pontibus,  a  Pocket-book  for  Bridge  Engineers.  .  i6mo,  morocco,  3  oo 

Specifications  for  Steel  Bridges i2mo,  i  25 

Wood's  Treatise  on  the  Theory  of  the  Construction  of  Bridges  and  Roofs .  .  8vo,  2  c :» 
Wright's  Designing  of  Draw-spans : 

Part  I.     Plate-girder  Draws 8vo,  2  50 

Part  II.     Riveted-truss  and  Pin-connected  Long-span  Draws 8vo,  2  50 

Two  parts  in  one  volume. , 8vo,  3  50 

6 


HYDRAbLICS.    " 

Bazin's  Experiments  upon  the  Contraction  of  the  Liquid  Vein  Issuing  from 

an  Orifice.     (Trautwine.) 8vo,  2  oo 

Bovey's  Treatise  on  Hydraulics 8vo,  5  oo 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

Diagrams  of  Mean  Velocity  of  Water  in  Open  Channels payer,  i  50 

Coffin's  Graphical  Solution  of  Hydraulic  Problems i6mo,  morocco,  2  50 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  oo 

Folwell's  Water-supply  Engineering 8vo,  4  oo 

Frizell's  Water-power 8vo,  5  o<^ 

Fuertes's  Water  and  Public  Health i2mo,  i  50 

Water-filtration  Works i2mo,  2  50 

Ganguillet  and  Kutter's  General  Formula  for  the  Uniform  Flow  of  Water  in 

Rivers  and  Other  Channels.     (Hering  and  Trau  vine.) 8vo  4  oo 

Hazen's  Filtration  of  Public  Water-supply 8vo,  3  oo 

Hazle hurst's  Towers  and  Tanks  for  Water-works 8vo,  2  50 

Herschel's  115  Experiments  on  the  Carrying  Capacity  of  Large,  Riveted,  Metal 

Conduits 8vo,  2  oo 

Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

8vo,  4  oo 

Merriman's  Treatise  on  Hydraulics 8vo,  5  oo 

*  Michie's  Elements  of  Analytical  Mechanics .8vo,  4  oo 

Schuyler's   Reservoirs  for  Irrigation,   Water-power,   and   Domestic   Water- 
supply Large  8vo,  5  oo 

**  Thomas  and  Watt's  Improvement  of  Rivers.     (Post,  440.  additional. ).4to,  6  oo 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  oo 

Wegmann's  Design  and  Construction  of  Dams 4to,  5  oo 

Water-supply  of  the  City  of  New  York  from  1658  to  1895 4to,  10  oo 

Wilson's  Irrigation  Engineering , Small  8vo,  4  oo 

Wolff's  Windmill  as  a  Prime  Mover , 8vo,  3  oo 

Wood's  Turbines > 8vo,  2  50 

Elements  of  Analytical  Mechanics , 8vo,  3  oo 

.      MATERIALS  OF  ENGINEERING. 

Baker's  Treatise  on  Masonry  Construction .  .8vo,  5  oo 

Roads  and  Pavements 8vo,  5  oo 

Black's  United  States  Public  Works Oblong  4to,  5  oo 

Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering 8vo,  7  50 

Byrne's  Highway  Construction 8vo,  5  oo 

Inspection  of  the  Materials  and  Workmanship  Employed  in  Construction. 

i6mo,  3  oo 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

Du  Bois's  Mechanics  of  Engineering.     VoL  I Small  4to,  7  50 

Johnson's  Materials  of  Construction Large  8vo,  6  oo 

Fowler's  Ordinary  Foundations 5 8vo,  3  50 

Keep's  Cast  Iron 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo,  7  50 

Marten's  Handbook  on  Testing  Materials.     (Henning.)     2  vols 8vo,  7  50 

Merrill's  Stones  for  Building  and  Decoration 8vo,  5  oo 

Merriman's  Text-book  on  the  Mechanics  of  Materials 8vo,  4  oo 

Strength  of  Materials i2mo,  i  oo 

iietcalf's  Steel.     A  Manual  for  Steel-users i2mo,  2  oo 

Patton's  Practical  Treatise  on  Foundations .. 8vo,  5  oo 

Richardson's  Modern  Asphalt  Pavements 8vo,  3  oo 

Richey's  Handbook  for  Superintendents  of  Construction iomo,  mor.,  4  oo 

Rockwell's  Roads  and  Pavements  in  France zamo,  i  25 

7 


•*  Lyons's  Treatise  on  Electromagnetic  Phenomena.   Vols.  I.  and  II.  8vo,  each,  6  oo 

*  Michie's  Elements  of  Wave  Motion  Relating  to  Sound  and  Light 8vo,  4  oo 

Niaudet's  Elementary  Treatise  on  Electric  Batteries.     (Fishback.) izmo,  2  50 

*  Rosenberg's  Electrical  Engineering.     (Haldane  Gee — Kinzbrunner.).  .  .8vo,  i  50 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     Vol.  1 8vo,  2  50 

Thurston's  Stationary  Steam-engines 8vo,  2  50 

*  Tillman's  Elementary  Lessons  in  Heat 8vo,  i  50 

Tory  and  Pitcher's  Manual  of  Laboratory  Physics Small  8vo,  2  oo 

Ulke's  Modern  Electrolytic  Copper  Refining 0 8vo,  3  oo 

LAW. 

*  Davis's  Elements  of  Law 8vo,  2  50 

*  Treatise  on  the  Military  Law  of  United  States 8vo,  7  oo 

*  Sheep,  7  50 

Manual  for  Courts-martial i6mo,  morocco,  i  50 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  oo 

Sheep,  6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo,  5  oo 

Sheep,  5  50 

Law  of  Contracts. 8vo,  3  oo 

Winthrop's  Abridgment  of  Military  Law I2mo,  2  50 

MANUFACTURES. 

Bernadou's  Smokeless  Powder — Nitro-cellulose  and  Theory  of  the  Cellulose 

Molecule i2mo,  2  50 

Bolland's  Iron  Founder i2mo,  2  50 

"  The  Iron  Founder,"  Supplement I2mo,  2  50 

Encyclopedia  of  Founding  and  Dictionary  of  Foundry  Terms  Used  in  the  • 

Practice  of  Moulding i2mo,  3  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

Eff rent's  Enzymes  and  their  Applications.     (Prescott.) 8vo,  3  oo 

Fitzgerald's  Boston  Machinist I2mo,  i  oo 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  i  oo 

Hopkin's  Oil-chemists*  Handbook 8vo,  3  oo 

Keep's  Cast  Iron 8vo,  2  50 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control Large  8vo,  7  50 

Matthews's  The  Textile  Fibres 8vo,  3  50 

Metcalf's  Steel.     A  Manual  for  Steel-users t i2mo,  2  oo 

Metcalfe's  Cost  of  Manufactures — And  tke  Administration  of  Workshops. 8vo,  5  oo 

Meyer's  Modern  Locomotive  Construction 4to,  10  oo 

Morse's  Calculations  used  in  Cane-sugar  Factories -. i6mo,  morocco,  i  50 

*  Reisig's  Guide  to  Piece-dyeing 8vo,  25  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  -3  oo 

Smith's  Press-working  of  Metals 8vo,  3  oo 

Spalding's  Hydraulic  Cement i2tno,  2  oo 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses.    . .  .  i6mo,  morocco,  3  p« 

Handbook  for  Sugar  Manufacturers  and  their  Chemists.  .  i6mo,  morocco,  2  oo 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  oo 

Thurston's  Manual  of  Steam-boilers,  their  Designs,  Construction  and  Opera- 
tion  8vo,  5  oo 

*  Walke's  Lectures  on  Explosives 8vo,  4  oo 

Ware's  Manufacture  of  Sugar.     (In  press.) 

West's  American  Foundry  Practice tamo,  2  50 

Moulder's  Text-book xarno,  2  50 

10 


Wolff's  Windmill  as  a  Prime  Mover 8vo,    3  oo 

Wood's  Rustless  Coatings:   Corrosion  and  Electrolysis  of  Iron  and  Steel.  .8vo,    4  oo 


MATHEMATICS. 

Baker's  Elliptic  Functions 8vo,  i  50 

*  Bass's  Elements  of  Differential  Calculus i2mo,  4  oo 

Briggs's  Elements  of  Plane  Analytic  Geometry 12010,  i  »oo 


Compton's  Manual  of  Logarithmic  Computations i2mo, 

Davis's  Introduction  to  the  Logic  of  Algebra 8vo, 

*  Dickson's  College  Algebra Large  i2mo, 

*  Introduction  to  the  Theory  of  Algebraic  Equations Large  i2mo, 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications 8vo, 

Halsted's  Elements  of  Geometry 8vo, 

Elementary  Synthetic  Geometry , 8vo, 


50 
50 
50 
*S 
50 
7S 
50 

Rational  Geometry 1 2mo,        75 

*t  Johnson's  (J.  B.)  Three-place  Logarithmic  Tables:   Vest-pocket  size. paper,        15 

100  copies  for    5  oo 

*  Mounted  on  heavy  cardboard,  8X 10  inches,        25 

10  copies  for    2  oo 

Johnson's  (W.  W.)  Elementary  Treatise  on  Differential  Calculus.  .Small  8vo,     ^  oo 
Johnson's  (W.  W.)  Elementary  Treatise  on  the  Integral  Calculus. Small  8vo,     i  50 

Johnson's  (W.  W.)  Curve  Tracing  in  Cartesian  Co-ordinates i2mo,     i  oo 

Johnson's  (W.  W.)  Treatise  on  Ordinary  and  Partial  Differential  Equations. 

Small  8vo,    3  50 
Johnson's  (W.  W.)  Theory  of  Errors  and  the  Method  of  Least  Squares.  i2ma,     i  50 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,    3  oo 

Laplace's  Philosophical  Essay  on  Probabilities.    (Truscott  and  Emory.).  i2mo,    2  oo 

*  Ludlow  and  Bass.     Elements  of  Trigonometry  and  Logarithmic  and  Other 

Tables 8vo,    3  oo 

Trigonometry  and  Tables  published  separately Each,    2  oo 

*  Ludlow's  Logarithmic  and  Trigonometric  Tables 8vo,    i  oo 

Maurer's  Technical  Mechanics 8. , ,    4  oo 

Merriman  and  Woodward's  Higher  Mathematics. 8vo,    5  oo 

Merriman's  Method  of  Least  Squares 8vo,     2  oo 

Rice  and  Johnson's  Elementary  Treatise  on  the  Differential  Calculus. .  Sm.  8vo,    3  oo 

Differential  and  Integral  Calculus.     2  vols.  in  one Small  8vo,    2  50 

Wood's  Elements  of  Co-ordinate  Geometry 8vo,    2  oo 

Trigonometry:  Analytical,  Plane,  and  Spherical i2mo,     i  oo 


MECHANICAL  ENGINEERING. 

MATERIALS  OF  ENGINEERING,  STEAM-ENGINES  AND  BOILERS. 

Bacon's  Forge  Practice i2mo,  i  50 

Baldwin's  Steam  Heating  for  Buildings i2mo,  2  50 

Barr's  Kinematics  of  Machinery ; 8vo,  2  50 

*  Bartlett's  Mechanical  Drawing 8vo,  3  oo 

*  "  "  "        Abridged  Ed 8vo,     150 

Benjamin's  Wrinkles  and  Recipes i2mo,    2  oo 

Carpenter's  Experimental  Engineering 8vo,    6  oo 

Heating  and  Ventilating  Buildings 8vo,    4  oo 

Cary's  Smoke  Suppression  in  Plants  using  Bituminous  Coal.     (In  Prepara- 
tion.) 

Clerk's  Gas  and  Oil  Engine Small  8vo,    4  oo 

Coolidge's  Manual  of  Drawing 8vo,  paper,     i  oo 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  En- 
gineers  Oblong  4to,    2  50 

11 


Cromwell's  Treatise  on  Toothed  Gearing 12010,  I  50 

Treatise  on  Belts  and  Pulleys i2mo,  i  50 

Durley's  Kinematics  of  Machines 8vo,  4  oo 

Flather's  Dynamometers  and  the  Measurement  of  Power. i2mo,  3  oo 

Rope  Driving i2mo,  2  oo 

Gill's  Gas  and  Fuel  Analysis  for  Engineers 12010,  i  25 

Hall's  Car  Lubrication i2mo,  i  oo 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Button's  The  Gas  Engine 8vo,  5  oo 

Jamison's  Mechanical  Drawing 8vo,  2  50 

Jones's  Machine  Design: 

Part  I.     Kinematics  of  Machinery 8vo,  i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

Kent's  Mechanical  Engineers'  Pocket-book i6mo,  morocco,  5  oo 

Kerr's  Power  and  Power  Transmission 8vo,  2  oo 

Leonard's  Machine  Shop,  Tools,  and  Methods.     (In  press.) 

Lorenz's  Modern  Refrigerating  Machinery.     (Pope,  Haven,  and  Dean.)     (In  press.) 

MacCord's  Kinematics;   or,  Practical  Mechanism 8vo,  5  oo 

Mechanical  Drawing 4to,  4  oo 

Velocity  Diagrams 8vo,  i  50 

Mahan's  Industrial  Drawing.     (Thompson.) .8vo,  3  50 

Poole*s  Calorific  Power  of  Fuels 8vo,  3  oo 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  oo 

Richard's  Compressed  Air 12 mo,  i   50 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  oo 

Smith's  Press-working  of  Metals 8vo,  3  oo 

Thurston's   Treatise   on   Friction  and   Lost   Work   in   Machinery   and   Mill 

Work 8vo,  3  oo 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics.  1 2 mo,  i  oo 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Weisbach's    Kinematics    and    the    Power    of    Transmission.     (Herrmann — 

Klein.) 8vo,  5  oo 

Machinery  of  Transmission  and  Governors.     (Herrmann — Klein.).  .8vo,  5  oo 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  oo 

Wood's  Turbines 8vo,  2  50 


MATERIALS   OF   ENGINEERING. 

Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering.    6th  Edition. 

Reset 8vo,  7  50 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

Johnson's  Materials  of  Construction 8vo,  6  oo 

Keep's  Cast  Iron 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo,  7  50 

Martens's  Handbook  on  Testing  Materials.     (Henning.) 8vo,  7  50 

Merriman's  Text-book  on  the  Mechanics  of  Materials 8vo,  4  oo 

Strength  of  Materials i2mo,  i  oo 

Metcalf's  Steel.     A  manual  for  Steel-users I2mo.  2  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Smith's  Materials  of  Machines I2mo,  i  oo 

Ihurston's  Materials  of  Engineering 3  vols.,  8vo,  8  oo 

Part  II.     Iron  and  Steel 8vo,  3  So 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

Text-book  of  the  Materials  of  Construction 8vo,  5  ot 

12 


Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials  and  an  Appendix  on 

the  Preservation  of  Timber 8vo,    2  oo 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,    3  oo 

Food's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

SteeL 8vo,    4  oo 


STEAM-ENGINES  AND  BOILERS. 


Berry's  Temperature-entropy  Diagram. 1 2  me ,  i  25 

Carnot's  Reflections  on  the  Motive  Power  of  Heat     (Thurston.) i2mo,  i  50 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book. . .  ,i6mo,  mor.,  5  oo 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  i  oo 

Goss's  Locomotive  Sparks 8vo,  2  oo 

Hemenway's  Indicator  Practice  and  Steam-engine  Economy i2mo,  2  oo 

Button's  Mechanical  Engineering  of  Power  Plants. 8vo,  5  oo 

Heat  and  Heat-engines 8vo,  5  oo 

Kent's  Steam  boiler  Economy 8vo,  4  oo 

Kneass's  Practice  and  Theory  of  the  Injector. 8vo,  i  50 

MacCord's  Slide-valves 8vo,  2  oo 

Meyer's  Modern  Locomotive  Construction 4to,  10  oo 

Peabody's  Manual  of  the  Steam-engine  Indicator I2mo.  i  50 

Tables  of  the  Properties  of  Saturated  Steam  and  Other  Vapors 8vo,  i  oo 

Thermodynamics  of  the  Steam-engine  and  Other  Heat-engines 8vo,  5  oo 

Valve-gears  for  Steam-engines 8vo,  2  50 

Peabody  and  Miller's  Steam-boilers 8vo,  4  oo 

Pray's  Twenty  Years  with  the  Indicator Large  8vo,  2  50 

Pupin's  Thermodynamics  of  Reversible  Cycles  in  Gases  and  Saturated  Vapors. 

(Osterberg.) i2mo,  i  25 

Reagan's  Locomotives:  Simple   Compound,  and  Electric i2mo,  2  50 

Rontgen's  Principles  of  Thermodynamics.     (Du  Bois.) 8vo,  5  oo 

Sinclair's  Locomotive  Engine  Running  and  'Management i2mo,  2  oo 

Smart's  Handbook  of  Engineering  Laboratory  Practice i2mo,  2  50 

%iow's  Steam-boiler  Practice 8vo,  3  oo 

Spangler's  Valve-gears 8vo,  2  50 

Notes  on  Thermodynamics I2mo,  i  oo 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo,  3  oo 

Thurston's  Handy  Tables 8vo.  i  50 

Manual  of  the  Steam-engine 2  vols.,  8vo,  10  oo 

Part  I.     History,  Structure,  and  Theory. 8vo,  6  oo 

Part  H.     Design,  Construction,  and  Operation. 8vo,  6  oo 

Handbook  of  Engine  and  Boiler  Trials,  and  the  Use  of  the  Indicator  and 

the  Prony  Brake 8vo,  5  oo 

Stationary  Steam-engines.  . .  .• 8vo,  2  50 

Steam-boiler  Explosions  in  Theory  and  in  Practice I2mo,  i  50 

Manual  of  Steam-boilers,  their  Designs,  Construction,  and  Operation 8vo,  5  oo 

Weisbach's  Heat:  Steam,  and  Steam-engines.     (Du  Bois.) 8vo,  5  oo 

Whitham's  Steam-engine  Design 8vo,  5  oo 

Wilson's  Treatise  on  Steam-boilers.     (Flather.) i6mo,  a  50 

Wood's  Thermodynamics,  Heat  Motors,  and  Refrigerating  Machines.  ..8vo,  4  «c 


MECHANICS  AND  MACHINERY. 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Chase's  The  Art  of  Pattern-making I2mo,  2  50 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

13 


Church's  Notes  and  Examples  in  Mechanics 8vo,  oo 

Compton's  First  Lessons  in  Metal-working izmo,  50 

Compton  and  De  Groodt's  The  Speed  Lathe izmo,  so 

Cromwell's  Treatise  on  Toothed  Gearing i2mo,  so 

Treatise  on  Belts  and  Pulleys i2mo,  50 

Dana's  Text-book  of  Elementary  Mechanics  for  Colleges  and  Schools.  .  i2mo,  50 

Dingey's  Machinery  Pattern  Making i2mo,  oo 

Dredge's  Record  of  the   Transportation  Exhibits  Building  of  the  World's 

Columbian  Exposition  of  1893 4to  half  morocco,  5  oo 

Du  Bois's  Elementary  Principles  of  Mechanics: 

Vol.      I.     Kinematics 8vo,  3  50 

Vol.    II.     Statics 8vo,  4  oo 

Vol.  III.     Kinetics 8vo,  3  50 

Mechanics  of  Engineering.     Vol.    I Small  4to,  7  30 

VoL  II Small  4to,  10  oo 

Durley's  Kinematics  of  Machines 8vo,  4  oo 

Fitzgerald's  Boston  Machinist i6mo,  i  oo 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  oo 

Rope  Driving i2mo,  2  oo 

Goss's  Locomotive  Sparks 8vo,  2  oo 

Hall's  Car  Lubrication i2mo,  i  oo 

Holly's  Art  of  Saw  Filing i8mo,  75 

James's  Kinematics  of  a  Point  and  the  Rational  Mechanics  of  a  Particle.  Sm.8vo,2  oo 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,  3  oo 

Johnson's  (L.  J.)  Statics  by  Graphic  and  Algebraic  Methods 8vo,  2  oo 

Jones's  Machine  Design: 

Part   I.     Kinematics  of  Machinery 8vo,  i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

Kerr's  Power  and  Power  Transmission 8vo,  2  oo 

Lanza's  Applied  Mechanics 8vo,  7  50 

Leonard's  Machine  Shop,  Tools,  and  Methods.     (In  press.) 

Lorenz's  Modern  Refrigerating  Machinery.      (Pope,  Haven,  and  Dean.)      (In  press.) 

MacCord's  Kinematics;  or,  Practical  Mechanism 8vo,  5  oo 

Velocity  Diagrams 8vo,  i  30 

Maurer's  Technical  Mechanics 8vo,  4  oo 

Merriman's  Text-book  on  the  Mechanics  of  Materials 8vo,  4  oo 

*  Elements  of  Mechanics i2mo,  i  oo 

*  Michie's  Elements  of  Analytical  Mechanics 8vo,  4  oo 

Reagan's  Locomotives:   Simple,  Compound,  and  Electric i2mo>  2  50 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  oo 

Richards's  Compressed  Air i2mo,  i  50 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     VoL  1 8vo,  2  50 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  oo 

Sinclair's  Locomotive-engine  Running  and  Management i2mo,  2  oo 

Smith's  (O.)  Press-working  of  Metals 8vo,  3  oo 

Smith's  (A.  W.)  Materials  of  Machines I2mo,  i  oo 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo,  3  oo 

Thurston's  Treatise  on  Friction  and  Lost  Yfork  in    Machinery  and    Mill 

Work 8vo,  3  oo 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics. 

i2mo,  i  oo 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Weisbach's  Kinematics  and  Power  of  Transmission.   (Herrmann — Klein.  ).8vo,  5  oo 

Machinery  of  Transmission  and  Governors.      (Herrmann — Klein.). 8vo,  5  oo 

Wood's  Elements  of  Analytical  Mechanics 8vo,  3  oo 

Principles  of  Elementary  Mechanics I2mo,  i  25 

Turbines 8vo,  2  50 

The  World's  Columbian  Exposition  of  1893 .4to,  i  oo 

14 


METALLURGY. 

e'gleston's  Metallurgy  of  Silver,  Gold,  and  Mercury: 

Vol.    I.     Silver 8vo,  7  So 

VoL  II.     Gold  and  Mercury 8vo,  7  SO 

**  Iles's  Lead-smelting.     (Postage  9  cents  additional) I2mo,  2  50 

Keep's  Cast  Iron 8vo,  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo,  i  50 

Le  Chatelier's  High-temperature  Measurements.  (Boudouard — Burgess.  )i2mo,  3  oo 

Metcalf's  Steel.     A  Manual  for  Steel-users-     i2mo,  2  oo 

Smith's  Materials  of  Machines I2mo,  i  oo 

Thurston's  Materials  of  Engineering.     In  Three  Parts 8vo,  8  oo 

Part    n.     Iron  and  SteeL 8vo,  3  50 

Part  HI.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  oo 

MINERALOGY. 

Barringer's  Description  of  Minerals  of  Commercial  Value.    Oblong,  morocco,  2  50 

Boyd's  Resources  of  Southwest  Virginia 8vo  3  oo 

Map  of  Southwest  Virignia Pocket-book  form.  2  oo 

Brush's  Manual  of  Determinative  Mineralogy.     (Penfi>ld.) 8vo,  4  oo 

Chester's  Catalogue  of  Minerals 8vo,  paper,  i  oo 

Cloth,  i  25 

Dictionary  of  the  Names  of  Minerals ...  .8vo,  3  50 

Dana's  System  of  Mineralogy Large  8vo,  half  leather,  12  50 

First  Appendix  to  Dana's  New  "  System  of  Mineralogy." Large  8vo,  i  oo 

Text-book  of  Mineralogy 8vo,  4  oo 

Minerals  and  How  to  Study  Them I2mo,  i  50 

Catalogue  of  American  Localities  of  Minerals Large  8vo,  i  oo 

Manual  of  Mineralogy  and  Petrography I2mo,  2  oo 

Douglas's  Untechnical  Addresses  on  Technical  Subjects i2mo,  i  oo 

Eakle's  Mineral  Tables 8vo,  i  25 

Egleston's  Catalogue  of  Minerals  and  Synonyms 8vo,  2  50 

Hussak's  The  Determination  of  Rock-forming  Minerals.    (Smith.)  .Small  8vo,  2  oo 

Merrill's  Non-metallic  Minerals:   Their  Occurrence  and  Uses 8vo,  4  oo 

*  Ptrnfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

8vo  paper,  o  50 
Roaeabusch's   Microscopical   Physiography   ot   the   Rock-making  Minerals. 

(Iddings.) 8vo.  5  oo 

*  Tillraan's  Text-book  of  Important  Minerals  and  Rocks .8vo.  2  oo 

WiUi&jns's  Manual  of  Litholpgy 8vo,  3  oo 

MINING. 

Beard's  Ventilation  of  Mines I2mo.  2  50 

Boyd's  Resources  of  Southwest  Virginia .8vo.  3  oo 

M«.p  of  Southwest  Virginia Pocket  book  form,  2  oo 

Douglao's  Untechnical  Addresses  on  Technical  Subjects  .  ....    i2mo,  i  oo 

*  Drioier's  Tunneling,  Explosive  Compounds,  and  Rock  Drills.   4to.hf.  mor..  25  oo 

Eissler's  Modern  High  Explosives 8vo.  4  oo 

Fowler's  Sewage  Works  Analyses I2tno  2  oo 

Goodyear's  Coal-mines  of  the  Western  Coast  of  the  United  States i2mo.  2  50 

Ihlseng's  Manual  of  Mining 8vo.  5  oo 

**  Iles's  Lead-smelting.     (Postage  gc.  additional) I2mo. 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe. .8vo, 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores .8vo. 

*  Walke's  Lectures  on  Explosives 8vo, 

Wilson's  Cyanide  Processes I2mo, 

Chl«  'nation  Process Z2mo, 

15 


Wilson's  HydrauLv  and  f  lacer  Mining I2mo,  2  oo 

Treatise  on  Practkal  and  Theoretical  Mine  Ventilation i2mo,  i  as 

SANITARY  SCIENCE. 

Folwell's  Sewerage.     (Designing,  Construction,  and  Maintenance.) 8vo,  3  oc 

Water-supply  Engineering 8vo,  4  oo 

Fuertes's  Water  and  Public  Health i2mo,  i  50 

Water-filtration  Works i2mo,  2  50 

Gerhard's  Guide  to  Sanitary  House-inspection i6mo,  i  oo 

Goodrich's  Economic  Disposal  of  Town's  Refuse Demy  8vo,  3  50 

Hazen's  Filtration  of  Public  Water-supplies 8vo,  3  oo 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control 8vo,  7  50 

Mason's  Water-supply.  (Considered  principally  from  a  Sanitary  Standpoint)  8vo,  4  oo 

Examination  of  Water.     (Chemical  and  Bacteriological.) i2mo,  i  23 

Merriman's  Elements  of  Sanitary  Engineering 8vo,  2  oo 

Ogden's  Sewer  Design i2mo,  2  oo 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis I2mo,  i  25 

*  Price's  Handbook  on  Sanitation i2mo,  i  50 

Richards's  Cost  of  Food.     A  Study  in  Dietaries i2mo,  i  oo 

Cost  of  Living  as  Modified  by  Sanitaiy  Science i2mo,  i  oo 

Richards  and  Woodman's  Air,  Water,  and  Food  from  a  Sanitary  Stand- 
point     8vo,  2  oo 

*  Richards  and  Williams's  The  Dietary  Computer 8vo,  i  50 

Rideal's  Sewage  and  Bacterial  Purification  of  Sewage 8vo,  3  50 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  oo 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) i2mo,  i  oo 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Woodhull's  Notes  on  Military  Hygiene i6mo,  i  50 

MISCELLANEOUS. 

De  Fursac's  Manual  of  Psychiatry.  (Rosanoff  and  Collins.).  . .  .Large  i2mo,  2  50 
Emmons's  Geological  Guide-book  of  the  Rocky  Mountain  Excursion  of  the 

International  Congress  of  Geologists Large  8vo,  i  50 

Ferrel's  Popular  Treatise  on  the  Winds 8vo.  4  oo 

Haines's  American  Railway  Management i2mo,  2  50 

Mott's  Composition,  Digestibility,  and  Nutritive  Value  of  Food.  Mounted  chart,  i  25 

Fallacy  of  the  Present  Theory  of  Sound i6mo,  i  oo 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute,  1824-1894.  .Small  8vo,  3  oo 

Rostoski's  Serum  Diagnosis.  (Bolduan.) i2mo,  i  oo 

Rotherham's  Emphasized  New  Testament Large  8vo,  2  oo 

Steel's  Treatise  on  the  Diseases  of  the  Dog 8vo,  3  50 

Totten's  Important  Question  in  Metrology 8vo,  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  oo 

Von  Behring's  Suppression  of  Tuberculosis.  (Bolduan.) i2mo,  i  oo 

Winslow's  Elements  of  Applied  Microscopy i2mo,  i  50 

Worcester  and  Atkinson.  Small  Hospitals,  Establishment  and  Maintenance; 

Suggestions  for  Hospital  Architecture :  Plans  for  Small  Hospital .  1 2  mo ,  125 

HEBREW  AND  CHALDEE  TEXT-BOOKS. 

Green's  Elementary  Hebrew  Grammar i2mo,  i  25 

Hebrew  Chrestomathy 8vo,  2  oo 

Gesenius's  Hebrew  and  Chaldee  Lexicon  to   the  Old  Testament  Scriptures. 

(Tregelles.) Small  4to,  half  morocco,  5  oo 

Lettens's  Hebrew  Bible 8vo,  2  25 

'.6 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 

This  book  is  DUE  on  the  last  date  stamped  below. 

Fine  schedule:  25  cents  on  first  day  overdue 

50  cents  on  fourth  day  overdue 
One  dollar  on  seventh  day  overdue. 


JAN  2 1  1948 


LD  21-100m-12,'46(A2012sl6)4120 


